When I start, why don't headlights work?

More than likely a bad headlight switch. check the fuses first, spray some WD40 into the switch second, and third; replace the headlight switch.

may be relay short circuit 2nd remote app problem

The new headlights, probably blew the fuse check the fuse and relay if so get a bigger fuse to run the new headlights. Remember when replacing any light in your vehicle to not touch the glass part of the bulb with your bare fingers..

One way to turn off the automatic headlights when the ignition is to apply the parking brake. That dinging sound you hear when you shift into drive is also common with the parking brake applied.

Release the parking brake.

Not a Corvette expert, but I don't think sunlight would have any effect.
The motors should close the doors when you turn the lights off regardless if it is dark or not.
Are the headlights staying on, or just the doors staying up ? The up position is the default when something is wrong, so I think it takes voltage to close them.

Your keyless entry system is going bad and you need a battery for your keyless entry system once you replace your battery with the new one thing should work properly

A simple jump start with battery cables should get the car started and recharge the battery. If you don't like dealing with battery cables, there are products that attach from inside the car using the cigarette lighter to jump start the car that work very well.

I love gremlins. ok first thing first. the headlights, you will see a fuse box in the engine compartment near the battery. remove the cover and you will find fuses and relays. if there is no label showing you which is which (look under the lid you just took off), feel them. the one very very hot is the one that controls the headlights (if they are stuck on a the time you do that.) replace that. it should sort out your headlight issue and possibly the interior lights.
now on to the interior lights. you haven't mentioned a blown fuse, i presume you did check them. same thing. check all the relays for one that is very hot, your door would be open anyways so the interior lights will be on at the time. replace that relay.
as a side note the headlight issue could be caused by the daytime running light module often bolted right to the firewall.
Those would be a good start and hopefully slay the gremlins

if its not un der the hood look under dash or open doors and check the ends of the dash where the door closes. yeah it was a dumb place to put it.