Suspect either a faulty gauge or sending unit in tank . Either way start by checking the wires on the fuel tank sending unit which also powers the fuel pump. Your looking for loose and or corroded connections.

Sounds like you have a ground problem on that light housing. Try running a jump wire from a known good ground to your socket or housing, if it lights up then you found your problem. Also does the truck have a trailer light plug? Those can be the source of many rear light problems as well.

You have to first determine weather it’s your parking lights or your brake lights, brake lights are much brighter than your parking (or tail) lights. When they’re stuck on go to front of car, look to see if the front or side lights are on as well. If they are then you have a stuck parking light issue if not and the rear red lights are only then you have a brake light issue. Either way first check that you have all the connections right on your new battery and starter and that no other wires were disturbed. If it’s ok and brake lights only then you can refine your question to brake lights stuck on. Which would probably be a brake light switch.

You can find this video on u-tube; if not a crank sensor specifically look for timing chain replacement.

You must remove 6 Phillips head screws which hold on the whole bottom cover. The battery door is part of this cover; and since you have a spare camera you can use the cover off of it. Also there are no wires and no soldering, but be careful not to mix up the screws they must be put back in the order they were removed….(note you need a precision screwdriver which you can get here at ifixit, and there very affordable).