Lets say I play a big chromatic chord (Like a tone cluster) - After hearing that chord, I have a tonic (Fact). My question is - will everybody get the same tonic?
No - some chords (like a major triad) do quite strongly point to a single root note (which might then be perceived as a tonic), but others (when you consider the superset of their harmonics) might not. Factors like how loudly the listener hears each note, and the timbre of each note, will play a part.
You also have to consider ear training. It may be that people with less 'trained' ears will be less likely to 'hear a tonic' (as they're less familiar with the concept), but then someone with a lot of musical experience might hear a major chord and also just think "that's a major chord - it doesn't establish any tonic on its own".
is the new tonal center 'choice' affected by the previous tonal center?
You'd imagine that there's a presumption towards the centre staying the same, or moving to a related centre, in the absence of evidence to the contrary. But once you're into chord progressions, all sorts of other factors come into play, like the timing and accenting of each chord - one could play exactly the same sequence of chords and elicit different ideas of the tonic from a listener. And again, different listeners will hear differently - some may be more given to 'hanging on' to a previously-established tonic in their minds.
Another factor is that a trained listener can choose to hear a tonic - if you play them a C major chord, they'll be perfectly capable of 'imagining' that the tonic is D, and hearing the C chord against that.