tls – What does the HTTPS server verify when it asks for a client certificate?

I assume that the only thing the server checks, is that wether the IP on the certificate matches the one for the current TCP/IP connection, …

That is very likely a thing the server does not check. Client certificates at least for HTTPS are usually not issued for a domain or IP, but for a user – because they are used to authenticate a user and not a system.

… that the certificate is signed by an authority the server trusts

This should be checked.

Additionally one might check for a specific subject of the certificate or the web application might decide based on the subject which user this is.