Yes, if you put the FOB in the tray the car will still start and hopefully after some miles driving it will charge. With the FOB battery discharged, none of the remote functions will work.
If it's dead when you put it in the tray it may take a few seconds to come to life and allow the car to unlock.
You can't bolt a raw panel to a car battery, you need some sort of charge controller. But don't overthink it. If your solar charger is any kind of quality, you really don't need to worry about it.
When the engine is cranking, the solar charger will "try to help", but its current is limited by panel output.
When the engine is running, the alternator ...
I would look into setting up a relay controlled by the ACC or RUN circuit. When the coil in the relay is not energized the contacts would connect the panel to the battery but when you switch the ignition on or to acc it would energize the coil and open the connection.
Typical 5 pin auto relay would look like below:
86 : ACC/ON
85 : GND
30 : BATTERY
Problem solved! (by my wife).
It turns out (a fruitless trip to the repair shop [the battery and alternator are fine] followed by a further episode of non-starting) that the issue was that I had 2 keys: an original and a copy. Both mechanically turn the ignition lock, but with the copy the ignition fires and then dies after a few attempts after which it ...
There are two possibilities, assuming of course that all the correct parts are fitted; compare dimensions and part numbers to anything removed from the car.
Either, the distributor isn't turning or the LT side of the system isn't triggering. First thing to do is turn the engine over with the distributor cap removed and make sure that the rotor arm is ...
The battery seems like it has plenty of power, so it it likely being charged properly. The click you are hearing is probably the starter solenoid (electronic switch) closing, but either it isn't sending power to the starter motor, or it is but the starter isn't working.
So it is most likely a bad starter motor, a bad starter solenoid or there could be ...