There's way too much under the Savvyhead to put in one love letter. All those questions are good questions, but you can't ask 6+ questions all at once -- pick a few of them, maybe, then ask the rest in play. Or, even better, figure out a love-letter move that will answer some of the questions implicitly.
Also, instead of having the 'first advance against the SWAT' be 2 weeks ago, I would suggest just starting the session with it: either don't use the move at all and start in the middle of the fight, or modify the move so that it provides a starting situation, then immediately go to that scene. And use that scene as a way to incorporate some of the questions you didn't ask outright, as well.
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For the Angel, don't just 'ask her where her loyalties lie and why' -- make a move about it! That's something you want to answer in an interesting way, not just through abstract conversation.
The point of moves in love-letters (and love letters in general) is mostly to generate exciting new situations for the PCs, not just to find out what they did in the downtime. The Angel's move to heal herself does not really do that -- I'd just let the Angel spend their kit supply to heal themselves in a straightforward manner -- except for the Shazza consequence. But it seems way better to have an interesting move devoted entirely to the Shazza possession -- and I mean a love-letter move, not a custom move (though you could have both.) Again, something that will both establish something about how the PC is dealing with the situation (i.e. give them choices as much as possible) and also create an interesting starting point for the session.
The Shazza move is a bit weird. What happens if the Angel doesn't try to get rid of Shazza? Also, it's kind of boring -- very generic results -- but that could be okay if Shazza is a well-fleshed-out NPC with not-boring desires. As long as Shazza getting that Hold seems exciting/interesting to the players.
Generally speaking, you don't need to have a rolled move for everything. Maybe this Tao kid just started working for her, full stop, no problem. Or maybe there are some problems, and the Angel has noticed them, but hasn't done anything yet. Maybe the move you want is just a simple choice about what she has been teaching Tao (pick one: first aid, rudimentary literacy, not to hang out with [bad people X], etc.)
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Your Brainer move doesn't 'give him a chance' to resolve anything: it forces him to resolve it in a particular way. On a 10+ he kills the guy... what if he doesn't want to kill the guy? The 7-9 results don't feel like very interesting choices, either, or at least not very Brainer-y choices. Where's the 10+ option to turn Barker into his psychic slave, or cut out his eyes, or do whatever other crazy apocalyptic thing the player might find interesting?
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The Gunlugger moves seem ok, but having both seems excessive. Having more than one important thing happen in a love letter seems like a bad idea to me, in general. If there's that much interesting stuff happening, you should be playing it out -- and again, the goal is to use the love letter to start something, not finish it.
I like the second move, but I would expand the list of choices and give them more options, for example:
'On a 7-9 choose 2:
* The target is definitely dead.
* You got paid.
* Whoever hired you is not pissed off.
* You didn't get hurt.
* The target/the target's friends don't know it was you.'
Or something like that. Don't just make them pick between 3 preset situations, let them mix and match a bit if possible. And in any case, make sure to start right after this went down, with the consequences still fresh and active.