What unlocks/perks persist to the next game
I'm unsure what carries over from game to game. I thought the defence grid bonus did as it says it is permanent, but after getting it to 30% before losing the game it started again at 15% on the next play through.
The only thing I have found so far is that you can keep one pilot from the game and take them into the next game with their XP/level in tact. Also once pilots are unlocked, you can choose which one to start with, but they start with zero XP.
Coins from achievements are also permanent, but they are only used for unlocking new Mechs, or is there something else they can be used for?
You pretty much covered it.
The pilot you choose at the end of the run can go with you with all their experience
any pilots you unlock are permanently available as the starting pilot choice.
any coins earned from achievements can be spent to unlock new mech parties to play with
certain achievements earn a permanent bonus, e.g. completing the first island allows you to pick any unlocked island as your first on a subsequent run.
Just like FTL, the game isn't designed around giving you massive advantages the more runs you played. Some mech parties may be easier to play with than others, reducing or increasing game difficulty. But in the end it is designed so that you can beat the entire campaign with your starting pilot and mechs.