Considering that a fixie hub can be built into any size rim, you can have a fixie in any size - as long as you're ready to pay for having a wheel built.
Depends entirely on what you think of as a "normal" bicycle wheel. An externally geared wheel is unlikely to work very well, as it has a longer axle and will require a wider dropout spacing than a fixie wheel. On top of that you might get into chainline and gear ratio issues, not to mention that the fixie chain might not sit comfortably on multi-gear sprockets. If you get the urge to turn the fixie into a geared bike, there are a bunch of brackets and attachments that your frame will be missing which will have to be sorted out. A wheel with internal gears is more like to fit both dropout width and chain width, but again using those gears will require some work. Replacing a fixie wheel with a single-speed wheel is usually easily doable, they tend to be a direct fit.
Any bike can be turned into a fixie, at varying degrees of difficulty. But if the rear wheel can be moved back & forth to adjust chain tension it'll be so much easier.
If it truly is a vintage bike - don't. Whatever value it has will disappear when you hack it up. But If you insist: Start with finding out what kind of dropouts you have. Only bikes with (semi) horizontal rear dropouts are good fixie candidates. Next check your rear OLD - Over Locknut Distance. Basically the width of your rear hub, the distance between the rear dropouts. Then go shopping for a fixie rear wheel with the same OLD. Install fixie wheel. Check what your chainline looks like. If you have a double crank, you might get a better (=more parallel with the plane of the frame) If you use the inner chainring. If that doesn't give a tall enough gear you might have to shift the rings around. Or swap to a Single-Speed crankset and a shorter Bottom Bracket. Once happy with that, fit chain and ride.
Don't think you can, unless you pay someone to build you a 24" wheel round a fixie hub. Fixed gear bikes are almost only used for road/track cycling, while 24" is MTB DH/ Dirt territory, where a fixie would be really awkward. Easiest kludge is probably to find a bike with 24" disc brake wheels, then slap on a Tomicog. I'll post a link about them. Remember that you need to ditch the rear derailer, and that the frame has to have horizontal dropouts.
Kinda-sorta. The wheel size has to match the frame/fork size, and the frame size should match the body size and riding type of the rider. Overall smaller wheels will be stronger and easier to bring up to speed than bigger wheels. But bigger wheels roll smoother and offer a higher available gearing - which rarely is important outside a race setting.
A wheel that has teeth that fit into the teeth of another wheel is called a gear.
The cast of Fixie Dave - 2008 includes: Dave Nice as himself
You need a rear wheel with a special type of hub (which can take a fixed sprocket), then you need a frame with (semi) horizontal dropouts. Put wheel in, adjust chain length and you're pretty much done.
Kinda-sorta. There are a couple of trikes, (like bigwheel) with the pedals stuck solidly to the front hub, these would be fixies, and for kids. Apart from that, fixies tend to be adult sized. It'd be entirely possible to build a kid-sized fixie, by simply building up a kid-sized rear wheel around a fixie hub. But I can't really see the point. A single-speed is almost as durable, and a lot easier to ride.
Fixie Dave - 2008 was released on: USA: 30 April 2008 (festival premiere)
That is a 'cog' wheel.