Two probably stupid questions

So I am kinda (super) new at the whole sex thing and am not on any sort of birth control on my end. A few nights ago, the day after my last period day, I had sex with my boyfriend and the condom slipped off in me. I am pretty sure there was some spillage. We got some Plan B the morning after. How freaked out should I be?

Also, not sure if anyone here would have the answer, but I’ll throw it out there. My boyfriend does not want children and is most likely going to have a vasectomy (I am okay with this). Does anybody know how effective that would be in preventing pregnancy (as in, is there a chance something goes haywire and I end up pregnant)? Would we still need to be using birth control/condoms if he got a vasectomy?