A 15 second wildfire wouldn't work with your other cooldowns. Overheat being 10 seconds doesn't help. Neither does 3 bullets which ensure you can only get double cleans no matter what. It'd end up also being a problem for reasons I'll list below

A 45 second wildfire won't work because you'll be desync'd from the rest of your cooldowns. BS, RF, REASS, and RICC are all minute cooldowns. 45 seconds means you don't get to use them again with wildfire until 3 minutes. In addition, you wouldn't even have bullets for every wildfire on top of having no CD's to amp the damage making it even more mild than our 3.x mild fires.

I do like the idea of putting more wildfires inside of our CD's to bring back our wild/mild dynamic, but it'll have to be done in a way that only takes 1 full loaded wildfire at a time instead of 2.

On the rest of your suggestions, a hotshot giving you 25 heat would probably end up being a problem if there's ever a situation where you can burn with flamethrower (like in V3S ironically) and you don't want to overheat so you can swap onto the boss, as cooldown below 50 heat is a waste. Changing how that works would also take away from the heat generation properties of FT and BS.

If I was to pitch in some actual suggestions that piggyback on yours
1) Make the 10 second cool down be removeable with barrel stab (but increase barrel stab back to 120).
OR
2) Make procs (and by proxy ammo) behave as if they were heated if you have ammo or proc a step below 50 heat.
3) Put more interactions inside of cooldown, increasing its potency the hotter the gun becomes, this would make a clear distinction between wild and mild. The break points shouldn't be 50 alone. It should be 25, 50, 75, Overheat, with an overheating cooldown doing the most damage.

And this is a bit of my own flavor but:
4) Provide a new type of reload that takes uses 50%-100% of your MP but provides either a free reload (pvp style) or a bullet that allows for a special debuff.
As i really think we need an ability to use our MP for something