Does your progress save?

Partly. What you earn in the lobby stays: gems, your class, unlocked maps and bots. What happens inside a run does not. If you leave a map halfway, that run starts over, cash and tools included.

Stays

Gems. Gamer Chad, 8:27
Maps you have unlocked. Baysic, 1:26
The bot you picked. German video, 0:27
Your class, picked in the lobby. Mykey Blox, 0:28
Lobby upgrades like bag size and walk speed. Bax, 0:43

Starts over

The cash you made in the run. Cos Wanderer, 4:56
The rake and blower you bought with cash in the shed. Cos Wanderer, 4:56
How much of each zone you cleaned. Cos Wanderer, 4:56

So if a run matters to you, finish the map before you leave.