If they track their DHCP Client IP, will I be caught?
Yes. Most routers themselves list the computers connected to them in the status panel.
If yes, how? By my computer name or IP address or MAC Address?
The router list usually shows IPs and MACs, and sometimes computer network names (which can be found out in other ways)
Of course, MACs can be spoofed so this is not hard to overcome.
If your computer is internet routable and didn’t use an anonymous network (of wifi proxies!?): Yes.
If your computer is not internet routable (it’s just a packet sniffer in stealth mode); or is using an truly anonymous network: Probably Not.
I vaguely recollect that WiFi security doesn’t include a client certificate trust chain by default. You can create a non-compliant WiFi NIC that pretends to be another node of the network.
EDIT: I think WiFi WPA2 and the like might be even less secure than SSL/TLS against spying. As man-in-the-middle attack in a wireless environment needs no cable to tap or ISPs to coerce. Sufficient distance/attenuation between two WiFi devices could allow for neither end to detect duplicate SSIDs.
I.e. From each victim node’s standpoint, attacker’s SSID is 1M distance away and the real peer’s SSID is 2M away, where M is the maximum reception distance – so the 2M duplicate SSID is never detected.