Do I affect the effect or the other way around?

The noun is usually "effect," and "affect" is usually the verb. As verbs, to "affect" is to influence and to "effect" is to bring about, as in "effect change," meaning create change.

Did I lay down or lie down?

You "lay" something. Annoyingly, "lay" also is the past tense of "lie." If you just took a nap, you say "I lay down for a bit."  Many people say "I laid down for a bit," but we usually reserve "laid" for when there is an object involved: "She laid the scissors on the mantel."

Is it ever OK to use "they" to refer to an individual

We do it all the time in speech. In writing, it would sure be useful, wouldn't it? You will find that sticklers stickle, but "they" as a singular, genderless pronoun is gaining acceptance.

Beside or besides? Toward or towards?

"Beside" means physically next to. "Besides" means "other than." They used to be interchangeable, but they've grown distinct. Most -ward words are interchangeable, but Americans tend to leave off the "s" while British English writers tend to leave it on.

If I'm talking about a cat, is it "awe cute"? "aww, cute"?

Awe traditionally is fear, respect, wonder. Awesome things are usually not fear-inducing these days, but as awesome as your cat may be, you are looking for "aw, cute." Some dictionaries accept "aww, cute" for particularly cute cats.