(1) Are sliding mitre saws a good investment for cutting laminate?

Yes.

(2) If so, would I need a special fine toothed blade?

Ideally, but it won't matter as the cut edge will be hidden under skirting.

(3) If so, can you recommend a saw? Cheapest at Screwfix (Titan 8") is
£79.99 and next cheapest (Erbauer 10") is £129.99. After that, it would be
cheaper to pay someone to lay the flooring!

I got mine from B&Q. They do a wide range of own brand Performance Power and
PP Pros. Good features to look for are things like twin metal slides and
255mm discs. Frivolities are things like laser sights.

Please remember to replace your skirtings. Those stick on beadings look
absolutely foul. The sliding mitre saw is essential to cut the new skirting.

Christian.