Why doesn't Blisk finish off Cooper after the Fold weapon knocks out BT with Cooper still inside, helpless? It's part of his philosophy. Cooper's wiped out most of his unit more-or-less single-handedly, so he's a Worthy Opponent. Blisk believes "You kill me, you're better. I kill you, I'm better", which implies that killing another person is a show of superiority to him. As such, how exactly does killing Cooper now, when he can't fight back, prove that Blisk is better than him? It's not a case of not killing him because he has no reason to (fiscally or otherwise), but a case of he won't kill Cooper in anything less than a fair fight, in order to prove that he's better than him.
Not just that, by killing the rest of the Apex Predators Cooper has ensured that Blisk is the sole recipient of their paycheck from Marder and the IMC for delivering the Ark core, instead of splitting it six ways had the rest of the team survived. Considering the high stakes of the Typhon operation, the contract was probably worth a lot of money, money he had just received moments earlier. For Blisk, a man who's explicitly Only in It for the Money, Cooper just did him a huge favour.
All his talk of "If you wanted me to kill him, it should have been in my contract" may have also been him being savvy, too. If he really wanted Cooper dead (which he logically would), he could have negotiated a new contract there-and-then to kill him. However, he possibly realized that sorting out payment with the IMC over Coopers' life would have taken so long that BT would have likely recovered by the time payment had been agreed upon, so didn't even attempt to, and instead opted to leave. Given what happened shortly afterwards, it was probably a good call.
And, by letting Cooper stop Marder's plans and destroy the Fold Weapon, Blisk will keep the Frontier War going, giving him more opportunities to ply his mercenary trade.
Another possibility is that Blisk is very much aware that his outfit is dead and if he wanted to take on another big contract, he'd need a new crew again and he doesn't want to waste obvious talent. He'll need people who've proven themselves effective and lethal... So, possibly that calling card is a sincere job offer to Cooper.
How exactly did Ash come back to lead the multiplayer function, given that she died inside an IMC facility that exploded, on a planet that also exploded? Sloan mentions finding both Ash and Kane before the planet explodes (more specifically, during the mission to repair the communications tower so the Militia troops on the planet can contact their forces safely), so its' possible they recovered Ash then, a good while before the planet explodes. Still, given the circumstances surrounding the two possibilities on how she died (crushed in BTs' hand, or being inside her Titan when it explodes), she can't have been in good shape when they found her.
The easiest answer is a form of Brain Uploading: as suggested in "The Art of Titanfall", technology has allowed the human mind to be inserted into robotic bodies as "Simulacrum", of which Ash and Spectre Pilots in multiplayer are classified as such. Whether this is a case of one's consciousness capable of being uploaded into any mechanical body or an actual human brain inside a chassis is not clear; however, the fact that Ash appears to lead the Vinson Dynamics faction for multiplayer, which is implied to occur post-Titanfall 2 campaign (since the Angel City map contains updated "Most Wanted" posters of Jack Cooper), suggests the former is likely.
According to the flavour text in Titanfall 2 and Titanfall: Frontline, Ash was rebuilt from a backup copy of her personality by Vinson Dynamics after the events of the campaign. However, not all of her memories were carried over, which is why she is seeking out the Remnant Fleet to retrieve them.