1) What is the normal airline all-engine acceleration height?
1,000 AFL
2) Does selection of FLCH or VNAV mode at that altitude provide sufficient rate of climb for terrain clearance or is V/S mode needed to get the aircraft climbing faster.
Yes. Climb power and +10 degrees pitch gives 1,000-2,000fpm depending on weight.
3) Do some airports require different procedures? For example Las Vegas.
Yes, and special pages are published for non-standard clean-up altitudes.
4) What altitude is the A/P normally engaged in the FLCH or VNAV mode?
Depends on the pilot. The lowest is around 10,000, while some fly up to cruise altitude. The average would be around 18,000.
This is what I see at my company. Techniques at other airlines may vary.