It does sound worrying, very much so. I would try to get your GP to explain as much as possible the reason for all this, there must be something they suspect, or several somethings, try to connect the dots.
Just my opinion but I would put off the lumbar puncture until last unless they have a compelling reason. My neurologist at Johns Hopkins offered me one as a sort of last resort after my other test results were in. When I asked if it would show anything or affect my treatment, which was none, he said not really. So they will do them without a clear benefit if they are just grasping in the dark. I would want more than that to submit to the procedure.