Hello, I currently have one BT router which pugs into my telephone line that provides internet to the rest of my house. I have another telephone line socket in another room, could I put another router plugged into that and provide a secondary network (or at least an alternative ethernet connection) from the same plan. And would the router have too be BT?
You can not run two routers of the one connection.
Would there be any way I could have an alternative ethernet connection without having to feed it all the way round my house from where my router currently is?
You could buy power line adaptors and use them.
See link
BT Broadband Extender Flex 600 Kit (084285) | BT Shop
You do not need to use BT ones and you can get cheaper ones if you shop around.
I've heard you can't open ports with those. Or is that unaffected?
No idea where you heard that from, powerline adapters just use your mains wiring as an Ethernet cable effectively.
TP Link have a good range https://www.tp-link.com/uk/home-networking/powerline/