Given you started smoking wires to the stereo, there's been some heat generated (likely shorted something).
So anything touching or near them could be heat damaged. And anything upstream/downstream of the current flow that generated the heat. Be it other swtiches, connectors, insulation melted, even the fuse box, and may be malfunctioning, or shorting which could cause more damage, even an electrical fire.
You need to identify where the stereo wires melting took place:
- physically, to see if the smoking/melting wires have damaged any wires, switches, devices, and
- everywhere those wires run to, looking for damage due to the high current that caused the high heat
But first, I'd recheck all fuses.
I'd unplug the new stereo from the car's harness, to take that out of the mix. Get the car working correctly first.
This may be something best left to someone knowledgeable, so you avoid the fire risk. Be it sitting or while driving down the road.
Not uncommon to be able to start an Eagle without the/a key. Less so with only 60 on it.