In order for the NO3 to give us N2 and H2O, wouldn't you need free H+
ions roaming around?

No. This process (denitrification) happens at an enzymatic/cellular
(anaerobic bacteria) level. The H+ come from general acids (protonated
amino acid residues, i.e. histidines) within the enzyme(s) involved.


Free hydrogen reacts pretty strongly with O2 if I
remember my chemistry correctly. Or does the free O-- released from the
NO3 pull the 2 H+ ions off the H2O and release the O2 molecule? But why
would the H+ ions do that?

Actually at the enzymatic level (nitrate reductase) it is OH- that forms
(not O--) and this quickly picks up a proton (H+) from neighboring general
acids to produce H2O.