If Nancy Pelosi was still the Speaker...

…would any budget cuts have been enacted? Would President Obama’s 2011 budget been accepted/approved by the House?

For Pelosi to be speaker, the Democrats would have had to retain their majority, so presumably yes.

However, the Senate would still have > 39 Republican members, so only that legislation which is ultra-right wing would have any prayer of passing.

Democrats run the government when they have a healthy majority in the House, at least 65 seats in the Senate, and the White House. For any other combination, the Republicans are in control.