Is something wrong with my router?

If you can, as already said, try to take homeplugs out of the equation. They are so variable I've found wifi to be more consistent. You can get external cabling, most decent places should sell rolls. If you wanted you could then put sockets in the rooms you need them and run the cable outside between them.

How easy is it to fit the connectors on the cable? I was thinking of getting the cable, with connectors already fitted.

Quite easy, can be a bit fiddly keeping the 8 wires in the right order when you push them into the plug. You just need a crimping tool and the connectors. You can have the coloured boots as well but they're not necessary.

If you wire them yourself you only need to drill holes for cables and not the connectors too.