"I'm guessing connection is made via an HDMI lead from computer to TV. Yes/no?
Or is it best to go for a DVI connection?"
If your PC and proposed TV both have HDMI ports then that's fine. The only difference between HDMI and DVI is that HDMI carries sound, DVI does not - you would need an additional audio cable in this case.
The only theoretical difference between a TV and a monitor is that a monitor can't natively display TV signals.
If you're happy that you won't want any resolution higher than 1080p then go for the TV. Monitors frequently support higher resolutions for gaming purposes.
TVs do as well, but you pay a lot more for them.
As for which TV to go for, that depends on budget and what other functionality you want in it. If you can provide more info someone I'm sure will make a suggestion!