Does fattas imply some sort of criminal activity or is the word interchangeable with saknas for instance?
No, "fattas" does not imply a criminal activity :). In the sentence above, "saknas" works as well, but the words are not always interchangeable.
Kvinnan saknas sen i måndags. This means that the woman is missing since last Monday (she "disappeared" last Monday) and here "saknas" cannot be replaced by "fattas".
Is there any reason to prefer fattas over saknas ever? I can passively recognise fattas but saknas is always what would first come to mind for me.
I recognise fatta as meaning get or grasp (and generally referring to an idea rather than something physical, when I've seen it used) . To be missing seems almost the opposite concept to that, unless of course somebody else took the thing, which I guess is why Mother was thinking some criminal activity might be implied. (What on earth is the etymology of fattas anyway? Is it really related to fatta?)
To me, "fattas" and "saknas" both sound good in the sentence above, but when it has to do with people there is a difference
Det fattas en person
Example: We need one more person (anyone) to be able to play a game
Det saknas en person
Example: NN hasn't turned up yet
By the way, "fatta" can mean a lot of things:
fatta ett beslut - take a decision
jag kan inte fatta det - I can't understand it
huset fattar eld - the house catches fire
han fattar tag i barnet - he takes hold of the child
In hungarian we often use the term "A wheel is missing/ His/Her wheel is missing " (Hiányzik egy kereke) for a lunatic person. Could you use the same in swedish. In english, I have never heard of it, but that's not my native.