Some stuff I read suggested that, when an ani-boot plays, it starts with the static "company logo" then plays the ani. Correct?

Incorrect. The company logo is animated, then it displays your custom static boot logo. Both can be replaced with Jemplode.

Just what logos and images can be replaced in the player?

A very clear itemized list of each of the things that can be replaced is here.

Is Tony's Logo Editor the only way to load static boot logos and some other statics? And is Jemplode is the only way to load ani-boots?

Short answer: For you, yes.

Long answer: if you know how to fiddle with memory addresses, command lines, and can build the file formats for these things yourself, you could do it by hand. But logoedit and Jemplode are the correct ways to do it.