How do you respond to ありがとうございます and other ways of giving thanks? Are there any expressions similar to the English "You're welcome" or "No problem", or is it appropriate to not respond at all?
The simplest one is 「いいえ」, "not at all". The next one up is 「どういたしまして」, "would do the same". Another one you may come across is 「とんでもない」, "don't mention it", or one of its more polite variants (replacing 「ありません」 or 「ございません」 as appropriate). There are even more polite responses, but as a 外国人 you will not be expected to have to worry about them.
It's the safe way to go but it' very long and can be viewed as a little too much for a simple "thank you" (this is still keigo)
Often pronounced as: いえいえ or いやいや as it is shorter
Japanese like to make sounds like this often. Can be used for very small things, like someone dropped a paper and you help picking it up. 「ん」with a smile works as well.
No problem. (To use with いいえ for clarity?)
Often preceded by いいえ -> That's ok, no problem.
Often preceded by いいえ -> Lit. "That was an easy task". Very polite.
Lit. "I'll do the same next time" Very polite.
Often preceded by いいえ -> No, that's nothing.
Say that to a friend. "oh, it's OK". Simple and widely used.
If you need to say thank you also (thank you too)
Lit. "No need to be polite".
Often preceded by うん or おー. "Don't worry, don't mention it"
No one said two really common expressions: どうってことない and いいってこと, followed usually by よ ... Also any similar expression ( どうってことはない, どうと言うこともない, どうって言うことはない and so on...).
To prove the use of ii tte koto, which I guess may be regional, here is a link
However, I'm not questioning how often you hear it in real life, I'm saying you hear it pretty often. And that's true if you watch television (you can read more in the comment).