Do BCP's work to protect against pregnancy regardless? For example, even if you are 'allergic', or if you have weird side effects, does it still work?

Not necessarily. The pill has a cited failure rate of about 5%/yr of use.Failure to follow proper instructions for use increases failure rate. You cannot be "allergic" to hormones in the pill that are modeled after the natural hormones of the human body.You can be intolerant of their side effects & when that leads to improper use they cannot work.