>What is UPT 5 cable? Is this what we call Cat 5 cable?

Yes. UTP (unshielded twisted pair) Category 5.

>My understanding is that DSL Router creates IP addresses for each PC connected to it. I think this IP is dynamic. Do I understand you correctly that the IP has to be static for the network to function?

On the outside, you will have one IP address, assigned by the ISP. On the inside, i.e. the side of your LAN, you will have addresses, preferably in the range reserved for private addresses (RFC 1918). The ranges for private addresses are 10.x.x.x, 172.16.x.x-172.31.x.x, 192.168.x.x. The addresses can be static (assigned directly on the equipment), or dynamic (on startup, the computer gets the address from a DHCP server, which may be the router). For a small network, I recommend static addresses - assigned on each PC - this should be easier to handle.

>When you are saying the solution with additional router. Are you saying to have two routers? Because my original plan was to have
> ...

The cable model has router capabilities, although it may be sold under a different names. So, I mean a router in addition to the cable modem or DSL-modem.

>And what is NAT/PAT service?

Network address translation / port address translation. The router should have this capability. So will a proxy.

In your internal network, you have private IP addresses, e.g. 10.0.0.2, 10.0.0.3, etc.; and some (often public) IP address assigned to your outside connection. The NAT/PAT server will "translate" these addresses back and forth. To allow the many-to-one translation which you require, port numbers (for the return address) are changed, too.