I bought a brand new set a while back will similar issues. Air them up and come back out an hour later to the bike sitting on the ground. Tried more air etc. with no luck. I ended up pulling the shrader (air) valve out and found that the o rings were faulty. Replaced the o rings and haven't had an issue since. It's air and the slightest hole will let them leak. Worth a shot. Hope it helps.
May have a clogged radiator, not getting enough circulation. Try a new radiator and clear the hoses. Completely drain the coolant and try it. The gear is working if its puking coolant.
Lights from Home Depot work just fine. They are regular halogen spot or flood lights. Depends on the model of trailtechs....some are harder to change do to their electrical connections on the back of the lights being hard wired. If its a newer style they will plug right in. Swapped bulbs out many of times.
The only way that bulb pictured would have been broken is from removing it from the housing. The lense is still intact and its in an aluminum housing?? Probably removed it and tried pulling the connection and broke the base off. Doubt that happened in shipping.