If you are breastfeeding then you should do the progestin only pill as the pill with the combo of estrogen and progestin has been shown to affect milk supply. It's called the mini pill. It's slightly less effective so you have to make sure that you don't miss any doses, as it will increase your chances of another pregnancy. They also have a progestin implant that is quite effective as well, and then you don't have to worry about if you miss a pill. And while a small amount of hormones do cross into the breast milk, there haven't been concern of it causing any issues with the baby. So I would guess you could start them as soon amounts your cycle regulates after having the baby.
Depends on the type of bc. I had the paragard inserted more than a month after I gave birth.