I mean, why is it then that in so many of the world's languages' terms for "mother" usually have an m phone or something similar?
Because this is one of the very first speech sounds babies make. Children acquire labials before coronals and coronals before velars. They produce voiced sounds before unvoiced and stops before fricatives or affricates. /a/ is the first vowel to be acquired because it has the most neutral vocal configuration--all the articulators basically at rest.
Which is why I think Georgian is kinda weird. Of course, there probably is a reason for the variation... I'll go look it up. It's probably some sort of etymological mix-up.
Do you also think that sounds from laughter and crying are always the same? That if, for instance, our ancestors' laughter would sound like momomo or something, we would laugh in that way? Or is the 'haha' thing the only way to laugh?
Actually, everybody's laugh is very different. Rarely do you find someone who actually laughs like "hahaha", unless they for some reason retaught themselves how to laugh. (I know someone like that... it's really creepy to listen to a real life [hahahaha:].) I think that's just an inaccurate transcription of the generic human laugh, since phones /h/ and /a/ seem to be the most common in said laughs.