Nihonto Club

-Do the smiths signature change over time?
Somewhat. The important books like Fujishiro sometimes show multiple examples by the same guy.

-Did the fakers have a copy of the authentic mei when they faked, or were they just winging it?
I doubt if anyone knows reliably. Some gimei are really close, and that would not be an accident.

-Are they supposed to look EXACTLY like the ones I see online on published shinsas, or is there room for error? My signature varies somewhat every time I write it.
There's definitely variation. When you have enough familiarity with kanji, you can see the handwriting features that tend to remain constant for one smith (usually).

-If the hamon, hada, jiri, and yasuri are the same as the smith in question, is that evidence for genuine mei?
To an expert, those things (plus more) are the ONLY evidence. Also add other features like shape, kissaki details, etc. The blade is studied with the signature covered up; you make a decision about who made it; then look at the signature. The signature by itself proves almost nothing, because in the end it's just chisel marks.

-Did the fakers 300 years ago just grab a random sword and fake the mie, or did they pick swords with the above characteristics to fake a name?
The faking was believed to be primarily for money reasons, so that leads to what you must do when making a fake. The idea (most likely) was NOT to fool an expert (which would be super difficult), but rather to fool someone with money, and someone who wasn't going to do their research. It's the same today with the gunto made in China. A lot of gimei were probably aimed at folks like merchants, who had money, but not expertise - they just wanted to claim they owned an important blade - nobody was likely to challenge them on it.

-How do I find out what type of hamon, hada, jiri and yasuri a certain smith used?
Other than the internet, it's a matter of checking as many of the respected reference books as possible. Most of those are in Japanese, unfortunately.
A good source is the old back issues of the NTHK and NBTHK magazines - they have English versions. Find out which issues have articles about the smiths you care about.

Keep in mind that "kantei" is still just an opinion. The big experts are NOT in the business of being uncertain, so their opinion will often sound absolute. On an unsigned blade, however, it is common to get an attribution to just a "school", because it's just not realistic to be too specific, when there might be a dozen smiths that did similar work.

Pete