"Random" tonal center
This will depend on several factors. As @Tim mentioned, if you have absolute pitch, then your conjecture (1) is correct, and any tune in C will remain in C for you, and likewise for the other 11 keys.
As for your conjecture (2), this is most likely, but it isn't nearly as random as you may think! Even if we avoid listening to music for an extended period, there are sounds all around us:
the buzz of AC circuits, also known as mains hum, power line hum (60 Hz in USA (~B half flat), 50 Hz in most of the rest of the world (~G half sharp)).
If you are in a more rural setting, you may hear birdsong (varies by species) and the chirping of insects (varies by ambient temperature!).
In a more urban or industrial setting, the rumbling of vehicles and
machinery will have characteristic frequencies as well.
The more ear training you have, the more your brain will take these sounds and contextualize them into pitches and intervals, and these will form the basis of the "random" tonal center in your head. For example: if you are in USA, you will hear a mains hum of 60 Hz in your office. Outside, a semi truck is idling (call it ~39 Hz for this scenario). These two notes are B and D# (roughly), producing a B major chord in your head. If you start humming while listening to those sounds, the tune will likely be in B major. Note that if you heard the exact same truck outside your UK office, your tonal center would be Eb major instead.
Conjecture (3) is mostly for non-musicians, who as far as I can tell by talking to them, somehow don't have songs in their heads at all times (??), which is really hard for me to even consider.