Do any national chess bodies not strictly follow FIDE rules? There are some subtle differences in the rules of chess used in different un/rated competitions. For example, chess.com never considers King + Bishop sufficient mating material (specific example here). 

Do all national chess bodies (e.g. AICF, FEDA etc) strictly adopt FIDE rules, or do some make their own variations like chess.com does? 

The more 'human' way of asking the same question is like this: Is chess played by exactly the same rules around the whole world, or are there subtle variations in the rules of the game from country to country, as is the case with some games/sports e.g. rugby union?
I do not know the rules for every country, but if the USCF and ICCF are any indication, most bodies have their own tweaks to tournament rules. The rules for the actual piece movements are consistent. 

The exception is when a tournament is both nationally-rated and FIDE-rated, then FIDE rules take precedence.