Acne
There is some limited evidence that dairy causes acne, but that may only be for some people (it could be different from person to person).
Taking a very small zinc supplement (around or below the RDA of 10mg a day) might help if you are low on zinc (or eating more zinc rich foods like pumpkin seeds).
However, effects of diet and even supplements are likely modest compared to medical intervention. Can you improve your symptoms a little with diet? Probably. Should you do that instead of taking medicine? No way.
B12 Related Acne
Only very large (unnecessarily large) supplementary doses of some vitamins and minerals pose health-risks.
In extreme doses, B12 supplementation can cause mild acne-like symptoms (if you experience those, it's a good idea to cut back a little). On more moderate doses, you will see no side effects.
Mild acne-like symptoms from large doses of B-12 (which go away when the dose is reduced) aren't really a health risk, but it's pretty well known and often mentioned in vegan nutrition guides.
Cyctic Acne
For cyctic acne, your best approach will be a medication like accutane (isotretinoin). A short course of treatment can be rough (dry skin, and some other annoying side effects), but it changes the structure of your pores and can provide long-term reduction of cystic acne. So, you should see a dermatologist for that (and talk to your doctor).
Although the branded medication "accutane" is no longer sold, this is due to the company pulling it voluntarily after a poor decision, without adequate evidence, by a jury costing the company money. But there are other generic versions available.
Regarding side effects: The only very severe side effect is birth defects, which is why regular pregnancy tests are required in some areas for women on it. Aside from that, the most severe side effect is a worsening of the acne before it gets better. Typical effects are dry lips and skin (need to use a lot of lotion and chap stick) and stuff like that. By no means fun, but the reduction in cystic acne is permanent for most people.
