Does that save my internet provider, my router, or my laptop? And how long are these stored?
DHCP is a protocol for distributing IP addresses in a network. This will, if you set it up, be done on your router, which will assign your computer an internal IP. Some routers also have a protocol about it, but not all. Depends as I said also from the settings that you make there.
That depends on which DHCP server you use. Mine stores under / var / lib / dhcp /. What you could mean by logs are most likely the leases. They are stored for as long as they are used, and a little more if a client wants to extend their lease. By default, no log is created for persistent storage, which could of course be done, but why?
Your ISP probably has a DHCP server for their network, but it's none of your business. He gives you no configuration. The address you get by PPP or similar. Your router probably has no hard drive to log on and your laptop will simply be a DHCP client that logs nothing but remembers its current lease.
Can an outsider theoretically hack into these servers and find out something about me?
Ah, thanks.
Can an outsider who is not physically close to me theoretically hack and intercept that data?
Theoretically, many things are possible, but the scenario for spying on DHCP is very special. The most exciting thing is the assignment of IP and MAC address. And since your internal IP address is one of the private ones that does not work on the Internet, an outsider can't do much with it.
Theoretically, that would be possible, yes. Depends on the technique you use and its administration