Only half a TMN, because you spec'd the 3.8 version. :-)Can I have a TMN on this
Normally that's what I would have done but I've not been allowed to drive for a while.I have the same problem so take the car for a run about once every 3-4 weeks to top the battery up.
Nope not true once the alternator spins it sees a potential difference in voltage between the battery & it's output & charges. A car will run without a battery connected but it's not recommended as battery acts a a smoothing shunt for the alternator outputThe alternator needs a supply from the battery. Generally there needs to be enough charge to illuminate the ignition light for a bump start to have any chance of working.
^^This^^Buy a new battery and fit it yourself. Its really simple.
Long term take your car out for long drives more frequently. Battery should be the least of your problems because its relatively cheap to fix ie you just replace it or charge. Leaving a car sitting for 4 months however will damage more expensive parts than the battery. Im thinking specifically engine seals that will not be getting lubricated and dry out, and the fuel will start absorbing water. But they are not the only things.
If you want to make the car last, use it.