Do soul trapped animals/creatures go to Soul Cairn?
Or is the reason why only black gems are forbidden, that only those send the souls to Soul Cairn?
And do all the souls captured with a (black) soul gem go to Soul Cairn?

It appears that they do because you can find animal souls in the Soul Cairn. Souls only go to the Soul Cairn when they are trapped in a black soul gem.

Thank you. I've yet to visit Soul Cairn so I didn't know.

Where do they go in normal Soul Gems? O.o

do the Souls even exist after using the soul gem?

That may be the creepiest realization I've ever had concerning the Elder Scrolls series...

Soul gems take only the soul's energy, this has been often discussed here in the forums.
So if they don't go to Soul Cairn they go to afterlife, probably Dreamsleeve.
And after my long studies I now think that souls in TES are bodys of minds
and while the mind stays in the afterlife, the soul gets recycled.
I might very well be wrong though, knowing how complex this is.

i assume after the soul in the gem has been used they are just sent to the dreamsleeve

Pretty much how I see it. We know that in the Dreamsleeve 'souls' are stripped of their 'memories and emotions.' Those stay, the soul goes back.