Can I erase some of my mix and match challenges that I don’t like or that have be given to me?

no

Nope, sorry- I do think like for the game of the month badges, those go away after that month is up, but you would have to not touch your gifts that whole time, but once you click on it the badges that are there are yours forever

No. It is one of the reasons I have pretty much stopped accepting any badges. I have asked Pogo to consider putting badges on a separate page so we can see what we are getting. That way, we can decide.