Waht was used for the sink to wall transition and as for the flanges they are not usually attached they jusy si there to cover the hole where the pipes come out if the sink is not attached properly and moves a bit then that might be causing the cracking but in the pic it looks like a really poor caulking job if they used silicone but if they used grout that was the wrong choice.

Thanks it looks like silicone to me not grout.. how can I repair It ?

And also i figured the flanges are supposed to be loose but they are almost too loose and sometimes hang down so thinking if i can put some sort of adhesive behind it. also how would i know if this sink wasnt installed correctly or not it looks pretty fine to me!

You could put a gob of something like museum wax behind the flanges--just enough to keep them from moving.

The sink photo is hard to decipher. Does the sink back against a painted wall?

Yes @Fori it is up against a painted wall! does that affect it?

Yes but honestly behind a sink tile is what oyu need since this is usually a spot whare water spashes can we see the whole wall My guess is the water is getting on the wall and somehow getting behind the material used for the space Remove what is there and do a backsplash with silicone in thta space beteen the tile and the sink and I usually use clear.

"it is up against a painted wall! does that affect it?"

Seasonal expansion and contraction will cause that unfortunately. Possibly may be latex caulk and not 100% silicone. I would remove it and try silicone as that tends be a bit more elastic and will move better,

Silicone caulk is unpaintable.

@millworkman can anyone tell me how should a pedestal sink be mounted to the wall? i just looked under and realize that one of the bolts is completely coming out from the dry wall and loose. clearly there is not a stud behind and that is maybe why its cracking because the right side isnt tight. could this be why? what can br done without a wall stud there for that bolt?