nav vIghajbe', ghItlhwI' vIghajbe' je
I don't have a pen nor do I have a paper

qovIjmey vIQejbe', ngavyaw'mey vIQejbe' je
I don't like small canine-like creatures nor do I like big ones

Is there a problem using {je} this way ?

No.

 I'm asking because it's
pretty tempting to translate these sentences as:

nav vIghajbe', ghItlhwI' vIghajbe' je
I don't have a paper, I don't have a pen too

qovIjmey vIQejbe', ngavyaw'mey vIQejbe' je
I don't love small canine-like creatures, I don't like big ones too

The last two fse translations "sound" strange for some reason which I
can't quite understand. On the other hand, I don't "feel" anything
wrong with the klingon ones.

In English, you would use either instead of too when the sentence is negative. Klingon je does not care whether the sentence is negative.

-- 
SuStel
http://trimboli.name