It can mean either, but it also has another meaning (at least in the US) that is probably more common. If I give someone a ride in my car, I would also say I picked them up. In that case, it has no romantic or sexual meaning at all. It just means I gave them a ride. Ex: I'll pick you up on the way to work. I'll pick you up in the morning. Pick me up for school in the morning. Another meaning can be to energize. In this case, dinner might pick you up, or a cup of coffee may pick you up.

In certain contexts, yes. That is what this phrase means. "We go to the bar to pick up women." Another good phrase the Americans use is "to hook up." This has a more sexual meaning than "picking someone up." "Jane and I hooked up last night."

I agree with Steve, though not many women would say they're going to bars to pick up guys.

it could be either one

http://www.englishbaby.com/lessons/5505/slang/pick_someone_up