When I choose keywords for my campaigns in AdWords, do I get ranked better for these keywords, or is this completely irrelevant?
Your question is not clear. I'm guessing you mean those keywords for your organic ranking. If so, Adwords does not help your organic rankings and vice-versa. They are two completely separate things.
What about content ads that google places on other websites? Doesn't google crawl those sites and see the links?
Having relevant keywords and ad text, a strong CTR on Google, and a high CPC bid will result in a higher position for your ad. Because this ranking system rewards well-targeted, relevant ads, you can't be locked out of the top position as you would be in a ranking system based solely on price. Also, the AdWords Discounter monitors your competition and automatically reduces your actual CPC so you pay the lowest price possible for your ad's position on the page.