Is it normal for a gallery to handle artwork carelessly?

No that's not normal and a pretty massive part of the job. Yikes.

Is it normal for communication to be bad, i.e. for an artist not to be able to reach the gallery with questions or concerns?

It can happen for legitimate reasons occasionally depending on what's going on. But if it's constant and you can never reach them, not normal and not good.

For a gallerist to ignore or completely miss emails/voicemails?

Similar to above, unless they have assistants managing that, not normal.

Do galleries care if artists in their roster are completely copying the motifs of other, more famous artists?

If there are a number of artists at the gallery doing that then yeah I wouldn't want to be on that roster, sounds like embarrassing company to be in and not great reputationally. No matter what scale it is, all in all it sounds like this gallery is not good. If you're just looking to make a living and they do a good job of selling your work, then that could be fine I guess. But if you're looking for a credible career doesn't sound like a good fit.