There exists a 50-ohm antenna covering the entire 3:1 frequency range of the AM broadcast band? That will fit on a car? Anyway...
You shouldn't need much gain on either band - most car antennas are totally passive and work fine.
Dynamic range and strong signal-environment performance is probably much more critical (driving close to in-band transmitter sites).
Matching in and out Z is going to be a head scratcher (I doubt anything is as simple as nice 50R or 300R).
Make sure you roll off the gain and keep it low up to the maximum frequency of your active device(s). If you use devices with gain up to microwave frequencies, then you need to pay attention to gain and feedback up to those frequencies (to prevent oscillation).
Also, don't forget to account for an added GSM mobile antenna installed a few inches from your antenna. You should probably include some components to filter out such frequencies.
Ground planes are a means to an end. They're not mandatory. If you end up deciding to use PCB with more than two layers, it might be cheaper to build two PCBs and stick them together (underside to underside) with a bit of glue.
Personally, I'd raid the local junk yard and see how others have done this. The Fuba antenna has about 20-odd components jammed into about 2 cc volume.