They're both okay and there's no difference in meaning, but the first one is used more often.

Quite similar. ‘from’ suggests a trip. ‘at’ suggests a place. I’ll pick my daughter up from the airport and meet the rest of you at Grandma’s. My job is to pick up pilots at the airport. (They’re there, of course)

They both convey that you are going to the airport and that somebody is expecting you there. But, I feel like there's a tiny tiny semantic difference. To me, when I hear "at the airport", I'm thinking that just means that the other person is there, they don't have to have taken a plane and landed there, maybe it's just a practical place to pick someone up or to park a car or to meet up. With "from the airport" I would assume that the person did indeed take a plane and land there and you might even go into the airport and wait for them. Now, the first scenario is quite unlikely and people will probably understand what you mean anyway, regardless of the preposition but if I ever had to make the distinction, that is probably what I would use ☜(⌒▽⌒)☞.

both are good

Yes, you can. "From" works as a direction, but "at" cannot: "I'm taking Bob from the airport to his hotel." (can't replace "from" with "at")