10A at 12V is only 120 Watts. Not much at all and that sounds entirely normal to me just running a few lights and some other electronic devices. Most auto batteries are in the 50 - 100 Ah capacity range which means that a typical one should be able to supply your 10A load for 5 to 10 hours. I think you will find that your battery has reached the end of its useful life.
Your title states "starter solenoid", yet in the question portion you're asking about the starter motor. Could you please clarify your question or title so we know what you're actually asking about? As described in the paper, the solenoid draws only 8-10A of power to hold it while it is starting an engine, yet can draw up to 25A to get things started (surge Amperage). The 35A it is talking ...
More often, a JSU does not have enough current supply capacity to provide the needed cranking current directly. Attempts to start the vehicle from a JSU that can supply say 10A when cranking currents are 100s of Amps will be completely unsuccessful.
Fairly straightforward question, would I be able to charge my car battery with a DC power supply that outputs 12V 2.5A? I know real car battery chargers put out 12V 10A. Would it just take 5 times
The fuse is for the entire circuit. There may be other things on that circuit, sharing that 20A fuse. Also, the gauge of wires used in a circuit is part of the "safe" current limits (maybe why your 12V outlet has a 10A limit), while the fuse is there more for current "spikes" rather than steady loads. I'd stick to the manufacturer's recommendation of a 120W steady load.