Does it matter which subject (Engineer, Doctor, or Lawyer) you choose in the beginning of the game and does this affect anything? Also if you choose to do the exam whenever you want then should I do it once I study enough or could I wait and unlock all the content with characters I can then take it?
The profession you choose, as of update 0.6.5, has no effect on gameplay, maybe aside from some VERY VERY minor dialogue. As of right now, all content is playable regardless of the choices you make, the profession you choose, the titles you assign etc... so don’t worry about that.
If I were you, I would choose the ‘Normal’ game mode, where you have a limited amount of time to study for the exam (20 days I believe)...the reason for this is because if you pass the exam on your FIRST try, TK has promised to add in a ‘bonus’ scene rewarding players who passed the exam on their FIRST attempt. Your 2nd question is a little confusing, but if you do in the end decide to do the ‘no time limit’ mode, then it won’t matter.