How can I monitor my kid's activities on TikTok?

There's no good way to do this. You can share an account with your child, so you're aware of the type of content TikTok might be serving them, but either way, you'll never really know for sure what they saw.

Should I let my kid make TikTok videos?

I wouldn't, but there's nothing necessarily wrong if you let them. Just keep in mind if their video goes viral, they could become well known by complete random strangers you have never met.

Can kids under 13 use TikTok?

Yes, there's a section of the app that puts even more restrictions on kids under the age of 13. The issue is that many kids will not like it because it's very restrictive, and it's easy to get around it by just putting in a false birthday.