I have a question. If you pick a conventional empire to start out with, does that mean all the NPRs in the game start out as a conventional empire as well to make it fair?
No.
Or do they automatically start out trans-newtonian no matter what?
Yes.
One more question, where can I find a topic or page concerning what constructions do exactly, how much I should have at once, etc? And by constructions I mean things like automated mines, mines, military academies, etc.
You can't, as everyone has their own idea about how many / what percentage is 'right'. Virtually every game of Aurora is spent rushing from 'too much of X' to 'not enough Y' to 'I need more Z and I need them yesterday'.
As for what they do, read the wiki, the FAQ forum, the Academy forum, the Mechanics forum, every AAR (after-action report) you can find, and make liberal use of the Search function. (Note: The search function is pretty bad, but it's better than nothing.)
Aaaaand concerning starting out one more time. How many research labs do you give yourself? I started with 20 in a conventional startup.
None. I don't mean I start with zero, I mean I don't "give" myself any. I start with however many Aurora assigns based on my population, conventional/TN decision, and government type, and go from there. Typically, this means three.
Of course, as soon as I've finished converting my conventional industry, I promptly build more - a LOT more. Usually so many more that I run out of available workers to man them all. Somewhere in the 30-50 range. Enough to employ all of my scientists.
More research facilities are generally my number one priority. The only things to rank higher are more mines (if my empire is running short of minerals) or automines (if a better mineral source than my homeworld is found, or my HW runs short of a critical mineral).