Research is helpful, but it's less important than any other factor on your application.
GPA and MCAT are the most important. As you know, your GPA is great. Make sure you do what you have to so you can kill the MCAT.
You also need to shadow physicians (your sister's colleagues are fine, but don't shadow your sister herself). If your sister can set you up with some primary care folks, that would be ideal -- that allows you to see a wider range of health issues and scenarios than you're likely to see with more specialized people. You also need both clinical and non-clinical volunteering activities. I'm not sure how the 200 hours you listed above breaks down, so you should look at that.
@LizzyM posted a while back about optimal time spent on various ECs:
How many volunteer hours are solid?.
And get some research if you can, but don't worry as much about it as you do about the other aspects of your application. Except for a few very research-heavy schools, med schools don't give research nearly as much weight as they do the factors I discussed above.
I wouldn't sweat the LOR thing. The professors who were willing to support your application to dental school should still be willing to support a med school application.
But the big problem is I think I have no chance as I have no research. This is always was my biggest reason of not even trying to apply as my pre-health advisor implied that.
My buddy
@Goro would tell you this just adds to his long, long list of reasons for believing most pre-health advisors are useless.