Battery read 10V dead, 14.5V while running after a jump, then dropped again once I switched off - which part is failing?

Take them one at a time, because each reading answers a different question and people run them together.

14.5V running tells you the alternator is producing. That is a genuine pass and you can stop suspecting it. It does not tell you the battery is accepting the charge, those are different things, and this is the confusion in what you have read.

11.5V after sitting is the important one. A healthy lead-acid battery at rest sits around 12.6-12.7V. 12.4 is about half charged. 11.5 is essentially flat, and getting there ten minutes after being charged means it is not holding what the alternator put in.

10V under the starter's load is then just the consequence.

So the shape of your evidence is: charging system fine, battery not retaining. That is a failing battery or something pulling it down overnight, which is your question 2.

The test that separates a dead battery from a parasitic drain, with just a multimeter:

Charge it fully, then leave it disconnected overnight. Take the negative lead off the battery entirely, note the resting voltage, and check it again in the morning.

• Still around 12.6V → the battery holds charge. Something in the car is draining it, and that is a current test next.
• Dropped to 11.5V again with nothing connected → the battery is the fault. Nothing else can be, because nothing else was attached.

This one test is worth more than a week of guessing and it needs no special equipment. Most people skip it and replace parts in order of price.

If it turns out to be a drain, the follow-up is a current measurement in series with the negative lead: but do that only after the overnight test, because half the time it is not needed.

Worth adding a common cause that fits your description exactly and is not a fault at all: a battery that was deeply discharged once and never properly recharged.

From the other thread on jump starts: driving does not fully recharge a flat battery, it only gets it most of the way. A battery that went flat, got jumped, and was topped up by short journeys will sit at 11.5-12.0V indefinitely and start the car on warm afternoons but not cold mornings. Which is precisely the pattern you describe.

So before condemning it, put it on a proper charger overnight and then do the disconnected test. If it holds 12.6 the next morning, the battery was never the problem: the recharging was.

That sequence also costs nothing and it saves a battery quite often.

Small thing that catches people mid-diagnosis: measure at the battery posts, not the clamps, and clean them first.

A corroded or loose clamp gives you a low reading that has nothing to do with the battery's state, and it also produces exactly your symptom - fine sometimes, dead others, depending on how the connection happened to settle.

If the numbers change when you wiggle the clamp, you have found it and none of the rest applies.