Position of pieces in square Is there a rule on how accurately pieces have to be positioned within a square? Can they touch the border for instance?
There are no FIDE laws governing these items for very obvious reasons.

FIDE does not legislate the size of the board, the squares or the pieces for ordinary play, nor the colour of the board nor the pieces. Since the pernickety days of Bobby Fischer these kind of things can be legislated for in something like the World Championship according to the demands of the players but it is unusual.

If you don't legislate for the size of the board, squares and pieces then it is nonsensical to start legislating for how pieces must be positioned within their squares. Obviously if you happen to be playing with kings whose diameter matches the width of a square then the king is going to be touching at least 2 sides of the square.

The FIDE laws of chess do give the following relevant guidance in the Preface:


  The Laws of Chess cannot cover all possible situations that may arise
  during a game, nor can they regulate all administrative questions.
  Where cases are not precisely regulated by an Article of the Laws, it
  should be possible to reach a correct decision by studying analogous
  situations which are discussed in the Laws. The Laws assume that
  arbiters have the necessary competence, sound judgement and absolute
  objectivity. Too detailed a rule might deprive the arbiter of his
  freedom of judgement and thus prevent him from finding a solution to a
  problem dictated by fairness, logic and special factors. FIDE appeals
  to all chess players and federations to accept this view.


The one line summary of that is not too hard to understand -

"Use your common sense!"

On your turn you are allowed to say "j'adoube" and adjust as many of the pieces as you like as may your opponent when it is his turn. However if you continue to do this repeatedly you may find that the arbiter comes and first warns you for disturbing other players and then starts penalizing you.

EDIT: For a real world example of pieces touching and crossing edges check out this Youtube video of a game between IM Basman and GM Fodor played in a FIDE rated rapid competition in August 2016 - https://www.youtube.com/watch?v=HYTy7xAT6uU .

The game starts just after the 3 minute mark on the video so start watching from about 2:45. Note the black c pawn which is 2/3 on c7 and 1/3 on c6. This doesn't disturb white who makes his first move without adjusting black's wayward pawn. It does mildly disturb black, though, who adjusts the pawn without saying j'adoube. His first move is b5 and the c pawn is adjusted en passant, as it were, with a backward nudge from his middle finger. Adjusting like this makes it clear that there is no intention to move the piece.

Now scroll forward to about 34:50 when white plays exd7. Note the white pawn on d7 is fractionally over the line into d8. Black plays Qxd7 and again the black queen is partially in d8. Both white and black players are unconcerned. Black plays Qb7 and again the queen is over the line and just edging into a7.

Scroll forward to the 40 minute mark and watch white play BxRc8. Note the bishop is just fractionally over the edge of the board. The black queen is still 90% in b7 and 10% in a7.

Both players, internationals in their own right, have no problems with this. 

Note that the Youtube channel belongs to Adam Raoof who runs these monthly FIDE rapid tournaments. He is a FIDE international arbiter who has officiated at World Candidate match level and is a qualified FIDE international organizer who makes a living running and organizing FIDE rated chess competitions.