On a generator, you have a prime mover (say, an engine) connected to the actual generator, which consists of either rotating coils of wire within a magnetic field, or rotating magnets surrounded by coils of wire.
The number of poles (magnetic poles) and the rotational speed determine the output frequency: Freq = Engine_RPM * Number_Of_Poles / 120.
Typically, a United States portable generator runs at 3600 RPM, with 2 poles, for a design frequency of 60Hz. Larger portable generators run at 1800 RPM with 4 poles here.
That is how frequency is determined. The number of turns and the magnetic structure determine how many volts are produced at the design frequency, voltage and frequency aren't related in any fashion except for design. Again, in the States, most portable generators are wound to have a 240VAC single phase output, which is center tapped and delivered as two 120VAC hots with one neutral, but virtually any voltage can be delivered.
The current output of a generator is determined by its load, as long as the load doesn't exceed the maximum capacity of the generator's prime mover (engine) plus the conversion losses of the actual generator. Prime mover power is often rated in horsepower (US) or kilowatts (everywhere else). With no losses, a 10 horsepower engine could deliver 7457 watts (actually VA for non-resistive loads) continuously, or 62.1 amps at 120VAC continuously. Try to take more, and the engine will slow down (reducing both the frequency and the voltage, which will also drop the current) until you reach a point that the engine actually stalls.
You get fluctuation of frequency and voltage as the load changes because the engine cannot respond immediately to the actual load change. There are regulators controlling the engine throttle that attempt to keep the engine at a fixed (design) speed, but it takes time for the engine to respond to new commands as it has to deal with varying fuel/air mixtures and combustion which aren't instantaneous.
As a clarification to other discussions here:
For a purely resistive load, halving the voltage would halve the current, and result in one quarter the power consumed. You can't say that just cutting the voltage in half cuts the power consumed in half. With some devices, that may be true, but it entirely depends on the load.
The factors that you are describing are just components of what matters here which is power (Instantaneous ability to do work or deliver power) and energy (work done or power x time acting).
You need a better understanding of power and energy and there is so much on the internet on this that repeating it here is not a good idea.
T address your questions, plus a few notes:
... to sell back to the grid.
To do this the equipment must be certified as suitable. Among other things it MUST be certified as "anti islanding" - ie it cannot send energy into the grid when the grid is unpowered. There is zero chance of getting approval for connection of DIY equipment to the gid without a certified anti-islanding arrangement from an approved supplier. The most usual and probably easiest method is to drive a certified ant-islanding "grid tie" inverter. These ost about $US1 per Watt on ebay for top quality brands (eg SMA) and rather less for brands of lesser repute. As a rule you don;t get what you don't pay for in this area. You can but non-certified grid tie inverters at low prices but connection is prohibited and probably also illegal. FWIW - the reasons fro requireing anti-isalnding are not overly good technically but rules are rules.
Which generates greater power: running at a faster RPM, or building a larger generator?
Essentially "No", although too small a generator for available power will limit output. .
Speed is not directly tied to power. It is often possible to build a more compact generator if it is DESIGNED to operate at higher speed but this is not "a given" and is complex. The power output is set by the available power to drive the generator as long as the generator is able to handle it. If eg a wind turbine receives 100 Watt of power at the shaft, you could design a slow or fast generator (or alternator) that worked about as well BUT you could only approach and not exceed 100 Watts no matter what speed you used.
Is there a limit on RPM usefulness in building a generator?
Yes,sort of - but aspects such as how is the power produced, are as great or greater influences in design speed. Most wind turbines rortate such that the blade tips exceed wind speed by a factor K. A k of 1 to 2 is slow, 2 to 6 say is normal, 6 to 9 is rather fast and 9+ is getting frenetic. Wind Turbines with TSR (Tip speed ratios wrt wind) of 12:1 exist but are noisy and prone to erosion by rain and dust. High speed devices fly apart more easily. A TSR = 12 machine is a work of art & insanity and I'd be wary of standing too near to one.
If I can make a generator run at 20,000 RPM (yes, I believe I could achieve that rotational speed), will that produce a useful current gain over the same generator running at, say, 3,600 RPM?
As above, it's essentially irrelevant, and 20,000 RPM is hard to design for well in that sort of environment - a rotor that rotates 300+ times every second is trying to kill itself and you every moment it is running.
I am also not sure if I should construct an AC generator, or a DC generator hooked to a DC to AC inverter.
AC is almost always used for reasons which become clear once you look at real world designs.
Best Answer
If the load is a small one, an inverter from 12V is the best way to go. In fact it's the only practical solution for a varying engine RPM source.
For a larger load, you need a separate generator unit with a fixed RPM.
That said, you can still buy 120V alternators that are belt-driven intended for under-hood generation. They were popular some decades ago on utility trucks as an alternative to a separate gasoline generator. Unfortunately they require a fixed RPM from the engine, so they aren't suitable for an RV in motion.
This company sells them: https://www.fabcopower.com/generat/bgen.htm