Hi. I'm online and happy to answer your question today.
It is OK to wear your contact in your right eye. One of the issues with your high myopia and the changes in the refraction after your cataract surgery is that you may be experiencing anisometropia with aniseikonia. This can cause the headaches, dizziness and nausea you have been experiencing. Here is a good article that explains what these fancy words actually mean:
Another thing that I have to worry about is that the pressure inside your operated eye, the one that had the laser treatment could be really high. Very high eye pressure can also cause all the symptoms you are having. The one thing that makes me think that this might not be the problem is that you notice your headaches and nausea are better when you wear an eye patch. If high pressure was causing your problems, an eye patch would not improve things. So. That's good news.
So. What is the take home message? I think you will be OK and you are doing the right thing by wearing the patch. Once your other eye has cataract surgery your eyes will be much closer to each other in prescription and this issue should go away.
By the way, how's the blob/floaters doing? Does your retina seem to be settling down?
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I see that you are offline. I'll switch over to the Q&A system. This system works a lot like 'text messaging' but an email is sent to each of us anytime something is posted to this thread. We can continue to work on your question there..... :)