Any entry to the UK by a non-EU national requires a visa/passport check (sorry, my wife who has permanent right to remain has the same thing).

Transfer depends on the time of day and whether you have a lot of luggage, but National Express coach is usually a good option. Train is more reliable for time (usually) but takes longer. The problem with road transfers is that you have the M25 and the upper section of the M23 which has roadworks now to contend with. Peak period is about 6.30 to 10, inside that period I would take the train. Outside National Express is probably OK. 3.35 is enough time usually.

There are lots of flights to AMS from Heathrow which you can do on a transfer basis (no visa) [EDIT - if you have checked bags you have to clear immigration and collect the bags but that's quicker than transferring to Gatwick]. May be worth trying to change to one of those.