Is it possible to get pregnant while on your period, but you took birth control pills to distort it without actually ever using them before and the condom breaks ?

Yes.

Yes, very easily.

First that is not the use of birth control pills, and that in itself can alter and disrupt, your monthly cycle for a month or two.

Even with condoms they have a high rate of failure, ( as you now know) so it is like not having one at all, since it broke.

While the chance is less, women do and can get pregnant while on their period.