Ian's rundown is pretty good. I'm learning simplified characters now because I'm going to work in mainland China, but Hong Kong and Taiwan still use traditional characters- same thing with a lot of Chinatowns in various countries. Ideally, I'd like to learn both systems; friends who know both have told me that there are a few rules for simplification that apply to most of the characters and that once you know those, it's not horribly difficult to switch between the two. I learned Japanese prior to studying Mandarin, and what surprised me was that some of the Japanese characters are identical to the simplified ones, while others are traditional. A Taiwanese friend said it's because some of the simplifications instituted in the PRC weren't totally new, just throwbacks to older simplifications that the Japanese took with them when they borrowed the Chinese writing system. I found that interesting.

You really need to write them out for them to stick, in my experience (I've studied Japanese before and am learning Mandarin now). Unfortunately, there isn't a way to completely avoid that. There are some great websites that make it a little less painful, though- I use Skritter (just Google it), and it's very good. If you have one of those tablets that lets you write on your computer, it works great. Other things that have helped me have been learning some of the common radicals and their meanings, which can help you tell similar-looking characters apart and sometimes come up with stories and ways to remember them. Once you've learned some of the common characters, just practicing reading can help with recognition, too. Try to avoid using pinyin if you can; the longer you rely on it, the harder it is to switch to characters, IMHO.

It's not bad, just very informal. You can use it amongst your friends if you're younger, but definitely don't call your boss "dude." If you're forty and calling people "dude," it could sound weird; it would make your English sound a little immature. It's not a swear word or something that's necessarily insulting, though. It's very North American English, I think; I've never heard a British person call someone "dude." And people from some areas of the U.S., at least, use it more than others.

To me, they mean pretty much the same thing. "That's it" is more flexible, though- it can come at the end of a conversation or discussion, sort of to say, "So, we agree," or "That's how things are," (this is when "That's it" and "That's that"), but you could also say, "That's it!" if someone guesses an answer correctly or does something right. If you said, "That's that!" in that context, it wouldn't make sense. I hope that's helpful- this is a pretty specific usage question.