What in the world is this TI component (bottom right corner)? It clearly has a TI mark on it but I've searched ti.com and elsewhere and I can't find anything on it.
The coil ( matsushita / Panasonic) is in the upper right 315Mhz.
The chip is a rolling code security chip to mate with the vehicle and send the appropriate digital signal sequence to the receiver in the vehicle. It is an application-specific chip and you will not likely find a replacement.
I don't know much about this stuff but I know enough to ask this: if I start poking around with my multimeter reading resistance in order to determine which pins go where, do I risk damaging anything (assuming it isn't damaged already)? There is only so much I can see.
I think the point is this is a customized chip and you will not be able to to purchase it.
You may be able to reverse engineer the circuit and find a microcontroller from TI that matches the pinout. But you will still need to program the chip to the right specifications which is no easy task.