A veritable butt-load of questions there. You are really asking for a Primer on airbrushing (maybe we'll write one someday), but for what its worth...
Airbrush:
Long answer: Airbrushes are either gravity feed or siphon feed, and either single action or dual action.
Gravity feed means the paint is in a cup and gravity pulls the paint down in. Siphon feed the paint is in a cup or jar with a straw and is drawn up into the gun by siphon action. Gravity works better, and is easier to clean. Only advatages with siphon is you can have a HUGE jar of paint, or several jars to swap color in a single paint session. For scale model work, hands down gravity feed is best.
Single action mean there is a trigger for air. The volume of paint is set by an adjustment at the back. With dual action the trigger press is for air while pulling back adjusts the paint volume. While at first single action seems attractive for a novice, you will very very quickly get frustrated by having to constantly stop to readjust the volume. Might as well just start with what you will inevitably end up with.
Short answer: Badger Patriot. Dual action gravity feed. Nozzle is a good general scale model use size, capable of some detail as well as laying down a base coat. Spare parts are everywhere. (Almost) everybody has one so getting help will be easy. Please note this is not the only good airbrush or airbrush brand in the world. There are many excellent options, but if Badger is the brand you have ready access to its a pretty obvious choice for a newcomer
Air Source:
You can use a hardware store compressor making sure to adjust the PSI waaaaaaay down to what airbrushes use. They are relatively inexpensive (small ones go on sale for like $60), but they are LOUD!
There are dozens of airbrush specific compressors on the market. While not silent they are quiet enough that they wont disturb, or in some cases will be barely heard by, someone somewhere else in the house. Be prepared to pay $200-$300 for a decent one with regulator (to adjust pressure) and moisture trap (to....trap moisture). Models with a tank are better than ones without.
A CO2 tank is another option (which can be expensive to get started and are not without certain cautions). DON'T even think about the disposable air cans, or using a spare car tire.
Paint:
This is one I really hesitate to comment on because its something people are VERY passionate about ("this is better than that" arguments seem inevitable), but especially since you mentioned noxious fumes, and you are new to airbrushing, I'll put Vallejo ModelAir paint out there as an obvious choice. They are ready to use right out of the bottle so no thinning required (which can be a finicky and un-precise science you can learn later). Moreover, they are totally solvent free. No noxious fumes, not even alcohol. Anything else will require an extracting paint booth to evacuate the fumes. With ModelAir you can, if you dont mind getting the room dirty, do without as the over-spray consists of just paint "dust" which will settle on every horizontal surface in the room. Wear a decent respirator with N95 or better while working though (even "non-toxic" paint dust doesn't belong in our lungs).