1988 Toyota Corolla won't start; clicking noise, no turnover

For the past week, car won't start when cold. When I turn the key, there is a loud "click" sound from the engine compartment that can be heard through the front firewall. The starter motor does not turn and, of course, the engine doesn't crank at all. Occasionally, the starter motor has turned over, and when it does, the car does start, but not always, and this occurs after many many tries and seems to be getting worse. Infrequently, the engine fan will go on. All the lights and dashboard lights work. But when the engine is warm, and then I turn off the engine, the car will start up again immediately. So is temperature part of the problem? My clutch pedal has been loose recently, so maybe that is it.

(Also, I'm having another weird electrical problem: every 6 months or so, when I back up, the back up light goes on, then off while in reverse. This lasts for a few weeks, then everything is fine for another 4-6 months. Is there some sort of lack of voltage in the engine? How does the Corolla 1.6 4AFE engine regulate voltage?)

But I am also thinking:
*starter solenoid
*starter
*low battery
*clutch switch
*ignition control wire

what should I check for first? Battery?