Does Android USB Shared connection runs a DHCP Server? I'm aware of USB Shared connection of android and I've been using it to give connection to my computer.

Also I've an old domestic router which has RJ45 and wireless interfaces. I've been thinking about connecting my phone directly to the router using an USB-RJ45 converter, in order to give internet to the computer based on the RJ45 switched interface of the router, plus some other wireless devices using WiFi.

So my question is: does Android USB Shared connection runs a DHCP Server?

Because in case it does, everything will run fine, but if it doesn't, I'm afraid it would only support one device to give it an IP.

I've found that in cyanogenmod does, but dunno in Android.
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.