Were other sins possible in the paradise? Except for not eating from the tree of knowledge were there any other things that Adam and Eve weren't allowed to do or was everything else allowed?
Adam and Eve were given only one restriction but they were also given a few instructions.  They were to "tend and dress" the garden, and they were to multiply and replenish the earth.  Failure to do these things would also have been contrary to the word of God and thus a sin.

Eating of the fruit of the tree of knowledge of good and evil was the only sin that carried a promised punishment; "in that day you shall surely die".

There existed, certainly, many other evil things that should not have been done in the garden.  Adam must not kill Eve.  But there was no need to give that sort of instruction since he had not yet been influenced by Satan and had not learned to kill, to lie, to hate, etc.  These sins would have been a possibility for you and I but not for Adam who was in a state of innocence.