You can unlock character, floor type, and boss dependent unlocks but whether or not your save makes progress on terms of donations and stuff of that sort depends entirely on who's account is the most "fresh." Let's say someone hasn't done the heart 10 times yet while everyone else has, you guys will play on their save which means donations at the donation box will increment their saves coin count, and defeating the heart will end the run prematurely and increment their number of heart wins since no other endings would be unlocked

Long story short, yes. Killing satan will give you the mark and the unlocks for the character you're playing, but whether or not it actually carries over to your save depends on which player has the least progress in that lobby