Can MP3 files have viruses? How can they be detected?

The answer to your question of Can MP3 files have viruses is that just like PDFs, websites, and other complex formats which need to be parsed, there could be parsing bugs. At best, such bugs trigger a warning or crash the program, and at worst (but not uncommonly) they are able to run code on your machine. Because it has a serious impact, these bugs are taken seriously and solved by the maker as soon as possible (for popular and supported projects at least). Having such a bug, which is widely applicable in case you want to hack someone, is quite valuable. If you use such an exploit on someone, you are basically giving something away which could be worth tens or hundreds of thousands of euros.

The risk of such a website having such an exploit and using it on you, and you having an unpatched instance, is probably negligible. It's more likely that they send you a file that is called something like music.mp3.exe which Windows sees as an executable .exe format and runs. Or some other bug that it not a bug in your mp3 player itself.