If to collaborate to make a new pet, do they both pay? And do they both get their own separate eggs?
Yes, when you hatch with someone you both pay. The cost depends on the level/pedigree of the pets, the better the pet the more you pay and you both get an egg. Even if you hatch adult pets of your own you pay. It can become quite an expense and at lower levels it can be difficult to have the amount of gold needed to hatch.