How can we check this still exist/that this is real?
You have to find some more information, the name of a Bitcoin exchange, some email messages from a Bitcoin exchange, some written notes.
A Bitcoin address usually looks like these examples
I understand what GUI wallet softwares are and why those are important for end users. But in a backend application, if I understand correctly, a wallet is simply a seed that can generate some private/public keys pairs in a deterministic way. Is that correct?
Correct, but you miss the most important feature of a wallet...crafting transaction / bitcoin script!...