You "lay" something. But, annoyingly, "lay" also is the past tense of "lie." Lay an object down. Lie down. He lay down. If you just took a nap, you say "I lay down for a bit" or "I decided to lie down."  If you say "I laid down for a bit," few would bat an eye, but we usually reserve "laid" for when there is an object involved: "She laid the scissors on the mantel."

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. Effect something and take the credit. Usually we talk about "effecting change," meaning you are creating the change. Affecting something means you don't create it, you just influence it.

We do it all the time in speech. In writing, it would sure be useful, wouldn't it? You will find that sticklers stickle, and it's still best to look for a way to write around the issue, but "they" as a singular, genderless pronoun is gaining acceptance.

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

It is, absolutely. Oh, the title to this section? "Questions I am frequently asked" sounds more natural, but my idiolect tends toward impish tweaking of both the rules and the counter-rules in the style of that thing Winston Churchill unfortunately never said about prepositions at the end of sentences. Once you know the "rules," you can play with them.

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 have relented and accept "aww, cute." Common is the variant "awww, cute." "Awwww, cute" even gets 421,000 Google hits. For more awesome cuteness, see this blog post from 2010:
http://markallenediting.com/2010/03/18/awe-spreads-faster-than-dictionaries-can-keep-up/