I want to run a web server from home
Does the ISP block incoming connections to my IP or is it all controlled from the router?
That depends on your ISP and also their terms and conditions:
It might be forbidden to run a web server at all.
They might allow it but ask you to upgrade your connection because of additional expected traffic or ask you to pay an administrative fee to open the ports.
They might be using Carrier Grade NAT which raises another set of technical issues.
If I buy a domain and want to use my computer as a server, am I going to be able to tell it the exact IP and Port to connect to?
If you have a static IP address then yes. Some ISPs will give you a static IP address for an additional fee.
If you don't have a static IP address then there are services like Dynamic DNS - Wikipedia. Some of these services are free (Free Dynamic DNS - No-IP.com - Managed DNS Services).
I do not believe port-forwarding port 80 is possible
Port 80 can usually be forwarded.
How could I setup a server on my computer?
There are plenty of tutorials. There is also a choice of software that can be used, depending on your technical ability.