If I needed to get from Heathrow to Vauxhall, I would never consider Heathrow Express. It takes you to a station nowhere near your destination, and doesn't save any meaningful time over the tube.
If going by tube, change from Piccadilly to District at Hammersmith, which is far far simpler than changing at South Kensington.
Oyster refund. You are correct that - at the moment - you can get the deposit back. It has leaked (in specialist media) that tfl intend to remove that feature during 2020. For the adult, using a contactless card or phone is better.
For the teenagers, speak to a staff member at a tube station (there are no ticket offices, but there will be a staff member roaming by the ticket machines) who will assist with young person discount.