What would be a minimum set of parameters that needs to be specified in order to avoid ambiguity in pairings?
The minimum would be to have just one rule. Do your first round pairings by sorting players according to their ID number... and then, in the second round, do it again. If you don't have any rules about repeating colors, scores, or meeting the same player twice, there's no ambiguity possible. If you don't want to flip a coin to see who gets White, just declare that the highest-rated player gets White.
This would, of course, be ridiculous and unfair. But it would be unambiguous and minimal!
Now if all of these rules/parameters have been set, is there still any ambiguity left? Or conversely, is it possible to satisfy all rules or would some get precedence in case they contradict each other?
Assuming you have sane rules (not like the one I described above), there are pretty much going to be rules which have precedence over other rules. If White wins more often than Black, there's no way you can have a color preference rule and a same-scores rule without one taking priority. Under FIDE Dutch rules, there is a minimum set of "absolute" rules, and many more "quality" rules with specified precedence.
For instance FIDE specifies that no player should play three games in a row with the same color. But if there are only two players (who both have played the last two rounds with the black pieces) with 7 points left who are therefore going to be paired. What is going to happen then? Is the FIDE rule of three same-color games going to be violated or are they going to be paired with other players first?
The following is considered an "absolute" rule:
non-topscorers with the same absolute colour preference shall not meet
Assuming that 7 points is a "topscorer" (has over half the maximum possible points), then pairing the people with the same absolute color preference would not violate the absolute criteria. And then, "minimize the Pairing Score Difference" has a higher priority than "minimize the number of topscorers or topscorers' opponents who get a colour difference higher than +2 or lower than -2" and "minimize the number of topscorers or topscorers' opponents who get the same colour three times in a row." So the answer is usually going to be that the rule of three same-colored games will be violated, so as to comply with the higher-priority rule that the pairing score differences be minimized. (It's possible that the various absolute rules would require another resolution, of course. For example, if the two players with 7 points had already played each other, they wouldn't be able to play each other again.)
Under the FIDE Dutch rules, with the exception of the coin flip at the beginning to determine colors, the pairings are very deterministic. They have rules like:
All the possible transpositions are sorted depending on the lexicographic value of their first N1 BSN(s), where N1 is the number of BSN(s) in S1
which means the organizers can't ordinarily decide things like which possible transposition to take. There's not any wiggle room.
The exception is if it's somehow impossible to comply with the absolute rules:
If it is impossible to complete a round-pairing, the arbiter shall decide what to do.
The only scenarios I could think of where this might happen is an extremely low turnout, or a mass walkout. (If you have 8 rounds and 8 players, you can't avoid playing the same opponent twice. Or, if everyone who played White quit before playing, after the first round everyone would ineligible for the bye because they'd already had a forfeit win, and after two rounds you'd also have non-top scorers that all had the same absolute color preference.)