Do they embalm prior to cremating?
Bad enough that he was in a coffin (for which I'm sure somebody paid a pretty penny for), much less having to be embalmed prior to cremation.
Damn morticians rape the living to bury the dead.
Ghouls.
Do they put someone in a coffin if they are going to be cremated?
Depends on state law. In Texas, a deceased has to be embalmed if not interred within 24 hours of death.
Now, if cremation occurs immediately, there is no requirement for embalming anywhere in the States.
If a family wants a viewing for a couple of days, embalming is a necessity, unless they want their friends to view a swollen, smelly mess.
My Dad had always complained about the same thing, he would say that he just wanted a simple pine box. He later decided on cremation rather than embalming and burial.
After he died, we went to the funeral home to make arrangements. When they showed us the coffin options for cremation, there was a simple plywood sheet with some sort of cardboard cover. Mom and us kids all looked at each other and said: "That's exactly what Dad would have chosen for himself!"
After his service was over, his brothers & sisters threw a big luncheon affair. That would have meant a lot more to Dad than some velvet lined maple box with brass handles.