I really like the PvP feature, it's really interesting. One minor comment though is that's it's easier to think in terms of how many points you have left instead of how many you have used, so you may want to think about displaying that number instead of how many total i have used.
The prestige system where you buy coal then buy heat is a bit weird, it would be more intuitive if you just got coal and the amount of coal you had increased money, then you would know how much you're actually prestiging for.
I would think about changing the prestige system so it's not tied to achievements, so it's possible to earn progressively more prestige to progress through the long hobby times. Maybe something like hearts earned sort of like other idle games, and then allow hearts to be earned even when the girl is maxed out.
Destroying satellites and asteroids gets obsolete basically right away in the game, I would suggest a way to turn minerals->money, and a way to scale up satellite money.
Know issue you need to refresh the game in order to reload it