Can you install Ethernet cables to at least one socket per room during full rewire?

Yes and yes. Both done at the same time.

You would need to run a seperate cat 6 cable to each ethernet outlet, the cables would need to be ran in a separate channel i.e they can't be ran side by side (touching). An electrician would be able to do this during the rewire.

Absolutely! Best time to do it when you're making all that mess already!

Yes, ethernet cables can be put near socket outlets and electricians can do this work. It can be run before, after or during a rewire, but it is more cost-effective to do it at the same time. There needs to be distance between the high voltage(power) cables and data cables, otherwise it can cause interference in the signal. When I do rewires, rather than wiring every room, I recomend wireing the places that benefit from a hardwired internet conection (e.g. TVs, games consoles and computers). Then we wire for Wireless Access Points, which are esentially wifi boosters that are hardwired, so reach much further than the router that your provider gives you.