(1) Now I want to setup a router in network A and give it one of these
static IPs.
(2) As far as I understand the PCs that will connect to that router
will get their IP automatically assigned and be able to use the
internet the same way users in network A are able to.
This is entirely workable. I do this myself.
Connect a LAN port of the added Router to Network A.
Turn DHCP OFF on the added Router.
You have said Network A is the DHCP Server (DHCP network A with static IP-addresses.).
So this way will work. It is one of several ways to hook things up, but this way works fine.
Alternately, you can hook up the WAN port of the added router to the network A (still static IP on Network A) and leave DHCP ON for the added router. Now it will be on a separate IP, separate network and still works. That is, the added network will have its own DHCP and IP address range.
One of the two approaches will fit your needs.