1) When changing from LGW-LHR you'll have to retrieve luggage yourself. It used to be possible to have it sent from LGW-LHR as you suggest but no longer.

2) Pass

3) Pass

4) Fairly straightforward. You can get a taxi (for about £70) which may be easiest if there are a group of you. You could take the National Express coach, they are fairly regular and go direct between the two. I've never used it to be quite honest but am guessing it runs to about £16-£20 per adult (I did look up the price once)

Alternatively if you are brave you could catch the train to Victoria from Gatwick then from there either take the Underground to LHR (take about a 45mins) or to Paddington where you can catch the Heathrow Express. For a group of three the extra to avoid all the hassle and take a cab may well be worth it.