Basal cell carcinoma biopsy possible if ingrown hairs?

I agree. Bcc is possible on any part of the skin. However, ingrown hairs are not a risk factor for bcc.
Ultraviolet exposure. Ultraviolet light exposure is one of the most important factor in the development of basal cell carcinoma. In grown hairs per se may not have a role in generation of lesions. Cysts of hair follicles may sometimes be associated with basal cell carcinoma.
Unclear question. I will try to answer, but I am not certain what you are asking. If you have an area with ingrown hairs and your doctor suspects a basal cell cancer is present there, as well, it can still certainly be biopsied, perhaps cutting out the area of ingrown hairs along with the bcc so as to solve that problem at the same time. I hope this helps.
Yes, not a problem. Yes, you can perform a biopsy in the vicinity of ingrown hairs. Do not let that deter you.
Two are unrelated. Basal cell carcinoma and ingrown hair are unrelated to each other and one does not affect the incidence of the other.