Yes, you can hatch two adult pets of your own. Just step onto the timer solo and the option will be given to you to hatch.
The talents the offspring has depends solely on the talents of it's parents. So, if the parents are suffering from bad talents, well, they'll be passed onto the offspring and likewise for good talents.
Keep an open mind when pet training as this process is very grueling, expensive and time consuming.
Not all the time. Let's say I had a Fire Cat and a Unicorn. They don't go together. So you'd get another Unicorn or Fire Cat. But let's say we have a Opossum and a Raging Bull. They go together so you'd get a Gallant Bull. But sometimes your 1st generation is better.
Q: How do you know if a pet goes with another certain pet?
A: W101 Wikipedia! Search up the pet and see at the bottom of the page if it can hatch with another one to create a 2nd generation! Good luck!