I want to hatch a pet with at least the talents Fire-Shot, Fire-Proof, and the appearance of a Heckhound. I don't care about the abilities. I have an Inferno Cat and Heckhound I'm planning to use as the parents. I also would like it if the pet could have Fire-Giver as a third talent (This is optional). Is this possible? Am I using parents that can make this possible? And if so, is there any advice for me before I dedicate the time and gold into this?
Yes, it's possible to get those talents transferred over with the hatching combination you're proposing. Both pets do have the talents you seek in their talent pool.
Talents that emerge after training have the best chance of being passed on through hatching. If you get fire shot as a teen or adult talent, for example, it would be best to stop training that pet right then and use it to hatch for the accuracy talent you want to transfer.
Hatching is an extremely frustrating, time consuming, gold eating activity. Plant gardens for pet snacks and to help increase your gold reserves. Cultivate patience and persistence, as this project will disappoint you at times.
One last thing to keep in mind, is that your heckhound has extremely low stats (strength, intellect, etc.). That will severely weaken whatever talents you do manage to get on there. The cat is a stronger pet in terms of stats, but not strong enough to pull the heckhound up into the useful range. To get the most out of a talent, you want all of your pet's stats to be at least 200 at minimum, and the higher the better.
Therefore, you might want to consider crossing your heckhound with a very strong pet to get the stats higher. Fire school pets are one great option - they usually have strong stats and unique fire talents.
Another option is to cross your heckhound with a Sea Dragon a couple of times - Sea Dragons have Spell Proof, Pain Giver & Sharp Shot, which are universal defense, universal damage, and universal accuracy. They also have Spritely, a healing talent. They're on sale this week in the crown shop, but you can also farm for them in Celestia. The Trial of the Spheres is where I got mine.