What, if anything, is an adverb?

My 7th-and-8th-grade English teacher Mr. Ohler knew his grammar, and approached it logically. I was thinking about that today when I read in Sports Illustrated an interview with former football player Lynn Swann. His mother didn't want him to play football, but when he snuck out and tried out, his mother said "If you make the team, you're not going to quit, even if you decide it was a mistake." Whoa! If you make a mistake, you should accept responsibility for it and all that, but often the rational, even wise, thing to do is to cut your losses by quitting. After all, you only live once, which I meantersay, life is too short to waste it following through on erroneous decisions. (That doesn't seem to be what other people mean when they say "You only live once", but it's what I mean here.)

Mr. Ohler had a thing about proper placement of the word "only". (Also about placement of punctuation marks before or after closing quotation marks, but I think he was on shakier ground there. And about starting sentences with "And". And for that matter, about sentences without verbs.) "You only live once" means that the only thing that you do once is live, which is true in its context, but rather vacuous, like saying "The universe is big." To express the intended meaning, I (and everyone else) should say "You live only once." But YLOO is less catchy and less pronounceable than YOLO.

But I've digressed wildly before there was even a hint of an original point.

I learned, no doubt from Mr. Ohler, that an adverb modifies a verb, an adjective, or another adverb. What about this:

"I don't want to play with or even against him."

Isn't that a case of an adverb ("even") modifying a preposition?

Or this:

"An adverb can modify a verb, adjective, adverb, preposition, or even noun."

Isn't "even" an adverb modifying a noun? "Even" is not an adjective or preposition, and it applies only to "noun", not any of the other nouns.

Mr. Ohler would have had an answer, but I don't know what it is.