No restrictions on ashes that I know of but little dog must be small enough to fit in carryon luggage that goes under the seat in front of your seat.
Not sure what your asking. Each airline has there own rules on animals. You need to contact them beforehand or you could be turned away. Most only allow a certain number of dogs per flight.. You'll need shot records as well. All shots must be up to date. Southwest is now $95. Always subject to change. No idea about ashes. I would call your airline and TSA. TSA is the one that should give you any problems.
Thanks! Your answer is awaiting moderation.