1. Why do all these transformers have so many pins?
Coil formers and their mounting parts are often made to cater for all eventualities, so several pins may not be used in a given transformer design. Sometimes transformers need several windings, ether to give several independent outputs, or to work together.in the same closed magnetics field.
Eg. power supply transformers for valve amps used to have a ~6.3V winding together with the 100+V winding(s), as this was always needed for the heaters.
2. How can I find what is the primary and what is the secondary?
If the transformers you have in mind are mains transformers with a fairly low secondary voltage (say 24V or less), it will be easy to measure with even the cheapest DVM - 200 Ohm range - lowest resistance pair will be the secondary and there will be no connection between primary and secondary.
Sometimes they're actually marked, or the PCB they were on were marked or could be traced to the mains input.
The wire gauge if can be seen will reveal it as well - the mains side is thin wire.
Some mains transformers will have several taps on the primary, to be used for 110V, 120V, 210V, 220V, 230V or something like that, often with a switch to set the voltage for the country the product goes to.
3. Is there usually more than one coil or more than one tap?
I just need to step up voltage for my coilgun.
Roll your own if you cannot figure one out.
Some transformers (mainly those found in SMPSU's) will come apart if you boil them for around 15..20 minutes and can be pulled apart when the epoxy is softened (don't use the pot for food after cooking epoxy though). Use heavy gloves to avoid burns.
In some cores the primary and secondary are on different sections and the one with thin wire should be spared while the other is cut and rewound (few windings for the low voltage side).
If you have a function generator and a 'scope, it's easy to tell the wires from each other - lacking the 'scope, a DVM with even a crude AC range can be used as well.