Error Codes: Prius only starts when jumped.

Start with the P0AA6-612. That's why the car doesn't want to start (except when it has just had the power disconnected long enough to forget it has a P0AA6-612, and hasn't rediscovered it yet). There is an electrical leakage path within the battery, which you may find responds to a good cleaning, costing you little more than some tedious labor.

Here's one thread (out of many you could search up) to get an idea what that may entail.

You may have seen a lot of threads suggesting that high-voltage gloves and safety warnings are overblown, working on the battery is no big deal, all the car's interlocks will protect you, and so on. A lot of that is kind of true when the battery does not have extra electrical leakage paths.

When it does and you know that's the reason you're working on it, because that's the code you have, the rules change. You do not know what voltages you will encounter where. Use the gloves.

Once that's sorted, you might look at the cooling fan issue. You can deal with the air conditioning at your leisure, if you care about the air conditioning.

Don't worry about the C1310; that's the braking computer acknowledging that there are trouble codes in the HV computer, and will go away once you have fixed those.