It has been proven that the minimum number of givens for a uniquely solvable classic 9x9 sudoku is 17.
The usual solving method followed by people is : Fill all the cells where you are sure there is only one option, using the rules of the grid and a set of methods. Edit : we will restrain the problem first to the single position method (linking this again) since this is what I have implemented in my c program :)
If you have not filled everything yet, make a guess. Then solve the grid, or encounter a contradiction : then backpropagate. Maybe making sometimes chained guesses.
1 (prequel) - Is it proved that some sudokus need you to guess once all the rules have been applied? From what does the need for a guess happen?
2 (question) - Is it proved that, if you have to guess, you can solve the grid only guessing for cells that have two options, and not more (not three, four ... nine options)? Or do some sudokus need 3-option guesses or more, possibly asking for the exploration of three or more sub-sudokus? Better said : are there sudoku puzzles with unique solutions that have 3+ options in every cell [corrected 11/17/2018 : replace by "3+ position candidate"] once the bare rules of the grid are applied?