Regarding the birth control thing, while I can’t give you specific scientific type advice, I do know there are tons of women doing hCG and plenty of them are on birth control, so in that respect I am not aware of it being a problem.
Also, Dr. Simeons, the one who made the original protocol, did not object to his patients using contraceptives while on the HCG diet protocol. You can download the original protocol here.
These days we do have a lot of different types of birth control than were around when Dr. Simeons was prescribing this diet, and one type of hCG birth control that does seem to be a problem for some is the Mirena IUD. I wasn’t on hCG when I had an IUD for a brief period of time, but it was a terrible experience that I feel caused my binge eating disorder – so whether you’re doing hCG or not, I’d research all the problems women with IUD’s have had it, it’s awful and so not worth it!