CO2 is cheapest but won't work on all metals. The wire itself. The wire is actually a tube with flux inside, the flux provides the shield. Some flux core wire is designed to be used with shielding gas. If the wire requires it it's necessary, for wire designed to be self shielding I doubt if it would hurt anything. I've never used this process but I believe most pull it. Solder melts at low temperatures, thus you will cause much less damage to the state of heat treating in the metal, less damage to any plating, dissimilar metals that might be difficult to weld can often be soldered, it's reasonably easy to do with simple equipment...
Bill

flec,
Welcome to the forum. I'll start with some simple answers....

Cheapest shielding gas is CO2.

Not sure what you mean by "functional electrode", .045 is very common.

The powdered "flux core" provides shielding gas (among other things), the slag also provides shielding.

Yes, depending on the electrode type, a shielding gas or combination of gasses can also be used in FCAW. This is refered to as gas-shielded (FCAW-G) versus self-shielded (FCAW-S).

If you add a shielding gas to a self-shielding electrode it can produce a defective weld.

A pull gun technique will give you a bit more penetration and a narrower bead. A push gun technique will give you a bit less penetration and a bead profile that is flat to slightly concave and and wider.