Car battery

Have you checked for an electrical draw? Have you checked the starter to make sure it isn't drawing too many amps?

Yes you need to have the CCA checked, the battery can have 12V but not enough amps to start.

If you check the CCA (cold cranking amps) make sure they are equal or greater than the cubic inch size of the engine.

I'd have the battery checked for free at one of the many places that do this as if it's the factory battery, 5 years is a good long life for them. Alternator sounds to be outputting fine

The alt does seem to be doing its job based on your first post. However, the battery may drip in voltage when you place it under a load.

Let me know what you find. Have it load tested to see if it holds up.