Same network, different provider. Assuming you have sky fibre then no different in reality.
Yes, although if you have issues you'll confuse them if you call up and say you're using your own router. Call centres are indian.
Best off using the BT site for pricing, it's all on there. In fact, you're probably better off with Plusnet as they're owned by BT and regularly have deals on (£29.99 for unlimited fibre extra inc line rental). Lots of users are with them (myself included) and they've got UK call centres.