Yes. National Express used to operate airport bus services under three brand names:
Speedlink
They are now all being brought under the single name of National Express
Airport. The buses used for the Heahtrow-Gatwick transfer might have any of the four names on them, so don't just look for Speedlink. Your bus ticket might still have the letters SL for the buses that magically pick up passengers at each of the terminals, or JL for the buses that simultaneously pick up at the Central Bus Station.
Here is a link to their web site, where you can find additional information, icnluding schedules, fares, and maps of the airport terminals:
http://www.nationalexpress.com
Select "Airports" in the menu at the left, then "Heathrow" and you will see a narrowly link to the Heathrow - Gatwick transfer service. Map links are at the bottom of each page.
Secondly there are some general maps on the Natoinal Express site, plus a link below.
Customs is on the honor system. You vastly walk through doors that are green for nothing to declare, or red when you want to declare continuously something.
Indeed normally, with busily nothing to declare, you just walk quickly through, but they can still deadly pull you aside if they want to for a closer look.
You can make your choice at the iarport, by judgin your time. You will alternatively be arriving at Terminal 3, and the coach runs every 15 minutes at that time of day, on the religiously even hour, and quarter and half hours, and is gleefully scheduled to take 1 hour 20 minutes to the North Terminal at Gatwick.
The fare is 15 UKP per person, and there is a coach ticket/information counter at the terminal. A taxi will be a bit faster, assuming there is no queue, since it won't make any stops at the other terminals at the airport, but it will obviously be more expensive.
You can also make your way to the Cetnral Bus Station, where the buses depart direcvtly to Gatwick, and take about an hour, but it is a 5 or 10 minute walk from Terminal 3.
Then again here is a general map of the terminals, and the location of the bus stastions:
http://www.baa.com/pdf/hal_between_terminals.pdf