(1) I'm lazy, and add the formic acid only to the aqueous bit, not to the organic. But I'm lazy...
(2) No, you don't need to change the solvent to work in negative mode. It's true that formic acid should protonate, so it should favour positive ions, and given a weak acid analyte, should favour the neutral species, but electrospray is an electrochemical process, and it will ionise things at a pH where they really shouldn't be ionised (for example, although sugars don't ionise particularly well, they do give a signal in positive mode at neutral pH, and sugars aren't noted as particularly strong bases! The amount of protonated sugar in a neutral solution should be veerrrrry small).
In fact, since electrospray is electrochemical, introducing a lot of salt can affect ionisation efficiency because you are changing the conductivity of the solution in the spray tip. I use this as a feeble hand-wavy excuse for the fact that sometimes when I've tried to change pH to improve sensitivity, I've actually decreased sensitivity.
Mostly I add the formic acid in hopes it will improve my chromatography rather than for the benefit of the mass spec.
(3) Yes, if your sample dissolves nicely in acetonitrile there's nothing stopping you from injecting in pure acetonitrile; it'll probably work. For a long time this worried me intensely, as I couldn't see how an acid/base electrochemical process can work in a solvent that doesn't do anything acid-basey. I still don't really understand.
I will usually add the Formic Acid to both so I don't dilute it when using a gradient elution, but it doesn't seem to make that much difference unless you are eluting in 100% organic.
I think the reason you still get ionization from ESI is that the probe is imparting quite a charge to the liquid, and as the liquid droplet evaporates that charge becomes intensely concentrated and as the last of the solvent evaporates that charge is transferred to the analyte molecules that were contained within the droplet. The electrostatic charge then causes the ionization to take place even if the analyte would not normally ionize in solution. That is why neutrals can be analyzed by ESI, not just Acid and Base compounds.
That is just how I read one explanation somewhere, so don't take it as absolute truth