I am pondering running a Cthulhu game.  There are two salient facts:  (a) CoC has an awesome setting and awful rules, and (b) I'm currently most familiar with the PbtA engine.  The Mythos World hack seems pretty straightforward: rename the stats, steal some moves from Apocalypse World and Dungeon World, figure out what to do with Sanity and Magic, copy the list of skills.  I'm not saying I've got it quite all figured out yet, but I think I've already got something I could run better than I could run by-the-book CoC. The thing I *don't* have is a decent system for money and wealth. 

I know I *don't* want everyone to have to know exactly how many dollars they have in their wallet, or in their bank account, or trust fund, or what their annual income is.  And since the broke private investigator and the dilletante heiress are both standard CoC characters, I know I need a system that's approximately equally easy to use for both.

I'm pretty sure that means the wealth rating has to be logarithmic.  (Just abstracting to "1-barter" or "1-profit" doesn't solve anything if one character feels lucky to have 1 or 2, while another has an allowance of thousands.)  For items below your current wealth rating, you can just acquire them without worrying about it.  Items above your current wealth are unavailable, or at least require extraordinary effort.  Items *at* your current wealth might require a roll, where one of the possible outcomes is that you exhaust your funds and have to reduce your wealth level.

It seems like this is likely to be a common issue across any games set in approximately modern worlds, from steampunk to space opera.  I don't want to go re-inventing it if someone else has already written it, but I'm far from familiar with the full range of hacks out there.  Can anyone direct me to a working system?

Lets say I tell everyone to roll Dicern Realities.

I think this is the heart of your problem.  As GM in Dungeon World I think you should hardly ever tell anyone, never mind "everyone", to roll Discern Realities.  It's really not a perception check like D&D.  If it's truly an ambush situation, *maybe*, but in a search-the-room situation like this, I definitely wouldn't.  You pick one player at a time, and say "What do you do?" -- either the one who's most keen to act, or the one who's gone longest without some speaking time.  Then if they want to search, they roll Discern, and you resolve the consequence before going to the next character.

But, you might say, what if they don't check for traps, and blunder into the room, and wham, there's a trap?  Well, if you're a kind DM, before they go in, you give some kind of hint of danger (bloodstains, skeleton, weird smell, whatever -- Show Signs of Impending Doom).  Then if the ignore your soft move, they've given you the golden opportunity for a hard move.  Or else, if you play this way, you could be within your rights to say, dude, you're in a dungeon, and it's a creepy statue room, and you didn't check for traps?  And go straight to the hard move.

But, you might say, what if there's a cool secret door or something and they walk past it and no one says "I search the hallway" and then my whole cool Fish Room subplot gets missed?  Well, one answer is, what of it?  Maybe you can re-use it later.  Another answer is, why are you hoarding your cool secrets?  If there's a clue someone could reasonably notice just walking casually by, just tell them, "hey, there's a cool, moist draft here, you can't tell quite where it's coming from."

But, you might say, what if it's genuinely an ambush situation, and we need to see if the party is truly surprised?  What I would do is pick one character with the best chance of noticing -- maybe the Ranger is explicitly scouting ahead, or maybe the Cleric is just the most perceptive -- and give her a chance to roll.  Success, and hey, you've noticed the goblins just in time, fail, and it's arrow to the knee time.  Roll damage.

It *is* worth thinking a little about the odds -- how many successful heals is the bard likely to get off before something goes wrong?  Let's assume you need a full success, because the GM can probably make sure that "unwanted attention" is distracting enough to keep the Bard from healing again right away.

With CHA +1:
0 heals: 28%  (failure on first roll)
1 heal: 52%  (partial success on first roll, or full success followed by a fail)
2 heals: 14.5%  (success followed by partial, or 2 successes followed by fail)
3 heals: 4%
4+ heals: 1.5%

With CHA +2:
0 heals: 17%  (failure on first roll)
1 heal: 49%  (partial success on first roll, or full success followed by a fail)
2 heals: 20%
3 heals: 8%
4+ heals: 6%

With CHA +3:
0 heals: 8%  (failure on first roll)
1 heal: 38%  (partial success on first roll, or full success followed by a fail)
2 heals: 22%
3 heals: 13%
4 heals: 8%
5+ heals: 11%

It's still not "infinite healing", but with CHA+3 the Bard does have a real chance of stringing a lot of them together, given enough time.  Which is another point:  the GM *is* allowed to make moves whenever she wants.  It's just that they should normally be soft moves if the dice haven't served up a failure yet.  But "Show Signs of Impending Doom" is a soft move, and can be used as often as you like to raise the stakes.  Maybe this apparently-safe place where they're resting up is starting to seem not so safe any more.

Besides mourning the previous companion, will the ranger be able to get a new companion? Of any creature or it has to be the same? How long does the new companion need to be trained to be able to do the same thing.

This is the kind of question where I think the Dungeon-World-style answer is for the GM to turn to the player and say, "Okay, Ranger, how does it work?  The rules don't really say, but you're The Ranger, so you should know.  What happens now?"

Then engage peer pressure around the table to make sure the answer is not lame.

As far as I can tell, that's the point of the "you're not A Wizard, you're The Wizard" stuff in the rules -- the point isn't that there aren't other people out there who might set things on fire with magic, but that if there are, they don't necessarily do it the same way you do.

There is an imbalance there.  Our party seems to have solved it by an unspoken "non-aggression" pact, where once someone has Cleric spell casting nobody else tries to take it because that would be stepping on their toes. 

Cleric and Wizard spellcasting and Druid shapeshifting are three cases where the heart of one class can be copied by a single multi class move taken by another.  You don't have to be ruthless in your min-maxing to notice that those are powerful moves. 

There are a few ways to rein this in if your players aren't the kind to do it on their own.  One is to fall back on the "to do it, you have to do it" rule -- make the players explain how they acquire something as amazing as the favour of a god, when they didn't have it last week.  The Cleric probably was born with it, or at least spent years as an acolyte.  What did the Paladin do, save that villager?  The whole party did that, why are you so special?  Play it out.  It could be awesome, and then it's all worth it.

Or else you can houserule in any number of ways, such as requiring more than one advanced move slot, or giving a penalty to spell casting if taken multiclass, or whatever.   But all my ideas here are pretty klugey, and I expect that's why they're not in the rules.

Dungeon World is not a wargame, and has serious flaws if you play it that way (what's the range on a Fireball, anyway?  What's up with Cloudkill?  Or Soul Gem? That's a weird spell.)  But it mostly works out if you focus on making the rules serve the story.  Either the broken thing makes no sense in the story, and therefore you can't do it, or else it pushes you to get creative about explaining it, and that makes it cool. 

But it is my job to make combat fun - to me that includes that the fights have to have some sort of tension to them. The characters don't have to be on the risk of dying all the time - but they have to fear the risk of dying every fight.

So, it sounds like you haven't actually run any Dungeon World combat yet.  One thing you'll learn is that the PCs are generally very powerful, and that you as DM will have to learn a few new tricks to make things challenging.  The example people new to DW usually point to and freak out about is that the standard dragon only has 16hp.  Even with 5 armor, if you let the PCs just stand there and wail on it, it goes down in a couple rounds. 

Making combat fun in DW is generally not about hit points.  For one, I think experience in other games is making you overvalue "whittling away hp" as a threat, and also to *undervalue* the -1 from gaining a debility.  A -1 in DW, rolling on 2d6 and looking to roll a 10+, is more like -4 or even -5 in a d20 game.  Plus, hit points are easy to recover, and debilities are hard.

Consider:  the Cleric and the Bard both have Cure spells that restore d8 hit points, and there is nothing in the rules to prevent a character from sitting there and just casting that spell over and over and over again after every encounter.  Note, I said nothing *in the rules* -- what prevents it is that eventually they will roll a failure.  It is your job as GM to give them a healthy fear of that failure.  Not so much fear that they just want to stay home, but enough that they think about it every time they pick up the dice.

So: how *do* you make combat interesting?  Cinematic tricks are good:  making fights about more than just standing and slugging.  Set fire to things.  Open cracks in the earth.  Make them run and gun.  Smart enemies are good:  have them threaten things that characters have to worry about.  (Ok, Paladin, do you hit the goblin in front of you and let the other two run past to tackle the princess, or do you block those two and let this one have a shot at your back?)

If you haven't done so, I highly recommend reading:

The Dungeon World Guide: https://dl.dropboxusercontent.com/u/3269630/dwdotcom/eon-guide/Dungeon%20World%20Guide%20pdf%20version%201.2.pdf
The 16hp Dragon: http://www.latorra.org/2012/05/15/a-16-hp-dragon/

This is all leaving aside another point:  other things players can do with those multiclass moves include:  Learn to cast spells like a Wizard;  Learn to cast spells like a Cleric;  Learn to shapeshift like a Druid;  Deal +1d4 damage (all the time) like a Merciless Fighter.  It's far from obvious to me that Bloody Aegis is even the most powerful option available.

Not really relevant, but in my game we never had a problem like this. We had this dialog on the first session:
-So what's easier to shapeshift into: small animals or big ones?
-Big ones, I have a very hard time shifting into small forms.
And then we played by this answer.

THIS.  Not specifically this, I mean, but the general principle of this. In DW when there's something vague, and a player thinks it's awesome and wants to do it all the time, and you think it's getting abusive, stop at some point and say hey dude, what's the downside of doing that all the time?  And they'll tell you, and you do that. Or else they tell you something lame, and you roll your eyes at them, and say, really?  That's lame. And then they give you something better.  Or else you ask the rest of the table (who might be also annoyed that dude is hogging the spotlight) what should happen. And someone will give you something really vindictive, and then you can be magnanimous GM and tone it down.

But make sure the player buys in that what you're doing is appropriate. I think that's key to being a fan.

Though also, I should point out that the Druid doesn't just get to transform automatically. He's gotta roll.  As GM, you owe it to the game to make players a little nervous any time they pick up the dice. Cause if they fail, you can make as hard a move as you like. ANYTHING could happen. That's when you bring in the hunting spiders, or make them forget being human for a while, or spring an ambush on the party (missing their Druid), or advance a Front so they start running out of time.