...i thought the war-captain would have a war company (judging by his name) but in the playbook there isn't a rule to allow him to have and create one (like the gangs in AW)...
I'm reading War Companies as temporary groups that disband before the end of a season and have to be mustered again. Or if you're mustering from different Peoples then it might reflect a temporary alliance that ends when they have completed a war campaign or, again, at the end of a season.
It doesn't explicitly say that the War-Captain starts with a War Company under their command, but I also don't see why you couldn't have the War-Captain be in command of one starting out - as long as it fits in with the fiction and doesn't artificially swell your People numbers.
I recently started watching
Vikings and
mustering troops is exactly what Ragnar does in the second episode, but by the fifth episode Ragnar is clearly no longer in command of the company as the men have all returned to their families and households are readying for winter... which leaves Ragnar vulnerable. If you're reading this then you should really put
Vikings on your list of things to watch!
...then the company gets its War from the majority or largest fraction.
I find the wording for that confusing. I think it needs to be more explicit because I'm just going to default to using the highest War rating and then maybe adding +1 if I think there are 30-59 warriors or +2 for 60+ warriors (because I can't get the number ratings for gangs in AW out of my head).