Hi Kristen. It's really sweet of you to do research for your sister and she's lucky to have you!
I don't know much about Lyme disease, but as far as I know, MS doesn't cause broken blood vessels. The only thing I really know about Lyme is that the symptoms are pretty similar to MS symptoms, and I think you have to find a really good lab to diagnose it. Many people are initially diagnosed with Lyme disease but find out later it's really MS, and vice versa.
I used to have a fainting problem, but my MS specialist says it is definitely not from MS. The other symptoms you listed are pretty common with it though. I'm glad your sister is seeing a neuro though; that's a great place to start. Good luck to you and your sister, and hopefully someone with more knowledge will pipe in soon. Hang in there!