Battery drain

No the starter will not drain the battery, have you load tested the battery to make sure it is okay, and what is the alternator charge rate, you can check that with a multi meter across the battery terminals set on 20 volt range, you want to see around 14 volt there about.

Or bad starter, bad alternator or bad battery will not usually cause a battery drain overnight. It should like you have a short circuit or something remaining on when the key is off. What you will need to do is disconnect the negative battery terminal, hook an ammeter up inline, then start removing the fuses one at a time, leave them out and check the ammeter after removing each fuse until it drops safely below 50 mA (0.050 amps). Then inspect that fuse's circuit for whatever does not work and why so hook the battery back up, put the fuse back in and this time use a voltmeter and find out where the item on the fuse's branch looses voltage and replace the shorted item or fix the short circuit, short to ground in the wire.

Yes it would because the battery would be drained. That is why the battery should always be tested and/or charged before condemning the starter.

Well, do you have the receipt from the work and any tests that were done before?
See if there is a starter draw test, a battery load test and/or even a mention of charging the battery.
Any short circuit can drain the battery overnight. The overhead light would have to be intermittently on while it is supposed to be off though in order to drain the battery. It is possible though that it is coming on intermittently because of the intermittently function that you did notice, using the ammeter would show you what is going on and when.

You are welcome and good luck!