Learn Polish: Ale tu nie ma żadnego śladu! - But there is no trace here!

No. In English, we use 'any' in negative sentences (like 'there isn't any trace'), but Polish uses a double negative. You must use the negative word żaden (none/not any) with a negative verb, not jakiś (some). So, nie ma żadnego śladu literally means 'there is not no trace.'

Why are the words żadnego and śladu in the genitive case here?

The phrase nie ma (when meaning 'there is no') always forces the noun that follows it into the genitive case. The base noun is ślad (trace), which becomes śladu, and the adjective-like word żaden must match its case, becoming żadnego.

I've also seen the word tutaj for 'here'. Why is tu used in this sentence?

Both tu and tutaj mean 'here' and you can use them interchangeably in almost all situations. tu is just a shorter, very common alternative.

Doesn't nie ma usually mean 'he/she/it does not have'?

Yes, when there is a specific subject. However, when used impersonally (with no specific subject), nie ma is the standard Polish way to say 'there isn't' or 'there is no'. It acts as the negative version of jest (there is).