Learn Dutch: Is dat jouw eerste hond? - Is that your first dog?

Just like in English, the word for "first" is irregular. Instead of adding the usual -de or -ste endings to the number een (one), the Dutch word completely changes to eerste.

Why do we use dat here? Isn't dog a de-word (de hond), which should take die?

You are right that hond is a de-word, but when you point to something and say "That is..." or "This is...", Dutch always uses the neutral words dat or dit. You would only use die if it was directly attached to the noun, as in die hond (that dog).

Could I use je instead of jouw in this sentence?

Yes, you can! Both jouw and je mean "your". Using jouw just puts a little extra emphasis on the owner—you are asking if this is your first dog, perhaps compared to someone else's. In everyday conversation, Is dat je eerste hond? is very common.

Why does the sentence start with the verb Is?