Does Android USB Shared connection runs a DHCP Server?
Yes. Definitely.
What a DHCP server basically does is not only dynamically allocating IP addresses, but auto-configuring some IP settings as well, for example DNS servicing.
Judging from a client's (let's say, your computer) behavior, it surely runs a DHCP server. If not, you'll have to configure IP settings (static IP) and enter DNS server addresses and Gateway address manually. But you don't have to do these by yourself, so you can confidently infer that a DHCP server is working.