To answer your questions:

1) no, bags will be checked to LHR, your family members will be responsible for collecting bags at LHR, taking them to LGW, and checking them back in.
2) i would say with bags as well it is the most convenient and cost effective, a private taxi would be even quicker and easier but at higher cost.
3) yes, that is fine short of a complete jam on the M25

Have a look at the LHR-LGW transfer guide in the sticky at the top of the board as well.