Yes, actually, there's a known connection between lactose intolerance and celiac disease.
With celiac disease, the immune system attacks the intestinal villi when gluten is ingested and damages them. Guess where lactase to digest lactose is made? The intestinal villi.
So when the villi are damaged, it affects the body's ability to make lactase, too. With some celiacs, when you have healed, you may regain the ability to have dairy again. Many of us never get it back, though.
However, to be prepared, it takes 6-24 MONTHS to heal from celiac disease damage, and that's if you never cheat and are very good on your diet AND are avoiding gluten cross-contamination. You may want to check out places like the forums at celiac.com to find out more about avoiding cross-contamination.
So, your symptoms - yeah, that's pretty much all stuff I've heard of celiacs complaining about before. It's definitely a possibility.
That said, some stuff you should know:
1. If your test is NEGATIVE, that means nothing in terms of gluten intolerance (although some doctors don't know this). There is NO test for gluten intolerance except for the elimination diet, so if the test is negative, that just means there's not celiac disease. The fact that you are feeling better on the diet is a good sign that there is at LEAST gluten intolerance.
2. If your test is positive, here's something that the majority of celiacs I know were never told by their doctors and had to find out on their own: because of the significantly higher likelihood that relatives of a celiac disease ALSO have this disease, and because this disease can be active for years with few to no outward symptoms, experts recommend that ALL family members of a newly diagnosed celiac get tested for this as well. And that they should get retested every 2-5 years for the rest of their lives, if it's negative.
3. If your test is positive and your doctor is not scheduling you for a lot of other tests, once again, hit the forums for celiacs. Celiacs have been shown to have pretty terrible after-diagnosis care, one of the worst for those with auto-immune diseases. But we still need it. We can be severely deficient in vitamins, but doctors often only check one or two- however the damage can be in different areas of the intestines, and some nutrients are absorbed only in certain areas, so you can have some vitamin levels that are fine, and some that are deficient.
We can have organ damage due to years of low nutrients.
We need to have our vitamin levels checked 6-12 months after we go gluten free, to ensure that we are healing right. And then have another celiac disease to ensure we are healing right (if it's negative, that means we've healed, as the test looks for damage FROM gluten, not something that is actually the disease itself). Over 50% of celiacs don't heal at first on their gluten free diet.
Recent studies suggest that this may be because a large number of them are getting gluten cross-contamination and aren't aware of it (it can give lower, barely there symptoms). Often, beginning celiacs simply avoid gluten ingredients and don't know that they need to avoid cross-contamination from factories and in their own kitchen, too.
On top of that, a small percentage of THOSE folks don't heal even when they eliminate all the sources of cross-contamination they know of. Turns out, ALL gluten free food has a level of allowed gluten contamination - this is an industry, with factories, so the ability to get completely gluten free is nearly impossible for them to achieve. This low level of gluten is tolerable for most celiacs, but a few DO react to it and require even MORE gluten free diets, with just fruits, veggies, whole meats, and so on (a recent article on this, actually:
http://celiacdisease.about.com/b/2013/03/04/study-trace-gluten-responsible-for-ongoing-celiac-symptoms.htm )
If you look at the studies, you'll notice that this last category is a LOW number of celiacs out of the total, something like 17 patients out of over 1,000. I only mention it because as you heal - over a few months, if you ever notice yourself NOT getting better, or hitting a wall, you may want to examine your diet, check for other intolerances (another common issue for celiacs who aren't feeling better), and you may have to do a lot of the legwork yourself and get any tests and help from your doctor second.
Wishing you good luck, truly. If you are like me, and have this...I cannot even DESCRIBE the difference it makes. You'll be amazed at how good you feel within a couple months. :-)