They can try but I’d imagine it will be filibustered. Democrats best shots at legislation will be via reconciliation, which is a senate budget trick that allows you to pass budget related bills with a simple majority (51 votes) instead of the typical super majority (60 votes). Republicans tried to replace Obamacare via reconciliation and failed, but were able to pass Trump’s tax bill with it.
The tricky thing is that there are arcane rules and limits on how to use it. Typically the senate passes one bill via reconciliation but you can do up to 2-3 I believe. I’d expect democrats to use one of those to fix Obamacare. They could use another for the $2000 checks.
Things that have nothing to do with the budget - like voting rights, qualified immunity, etc - can’t be passed thru reconciliation. Meaning they will need nine republicans to join them to pass that stuff. Unlikely...