What does the word "own" mean in this sentence? Is it ok to omit "own" here and say "in our cat's eyes"?

What would be good alternatives that could replace the word "tip?" If there are any, could you give me some examples?

Althought "tip" is the best word for this situation and any other situation that requires the giving of small pieces of advice, you could potentially use "piece of advice" or "trick" to replace it. Ex: I can share some cooking tricks with you. Ex: Do you have any pieces of advice to share?

What does "stamp" mean? Is it used in general conversation?

To "stamp" in this context describes the action of raising your foot up above the ground and bringing it down hard and loudly to hit it on the ground. This meaning of the word "stamp" is not used very often, unless you are specifically describing this action. In American English, many people say "stomp" instead of "stamp". Both words have the same meaning. Ex: The child stamped his foot in defiance. Ex: Stop stomping! You'll wake up the baby.

What's "fly" mean when used as a noun like here?

Here, the noun "fly" means the zip or buttons of one's pants at the crotch. Ex: It would be very embarrassing to have your fly down on a runway. Ex: Zip up your fly!

What does "your highness" mean?

Your highness is used to address someone in the Royal family.

What does "give it a try" mean? Should I always put "a" when I use this expression?

The term "give it a try" is an idiom and therefore is always in this structure. While "it" can be replaced by the specific thing you are talking about, the rest of the phrase must remain the same. To "give something a try" means you will try to do something. Mostly it is used when someone gives you advice on how to do something or a suggestion and they are encouraging you to try. Ex: I always wanted to give windsurfing a try. Ex: It's a little spicy but give it a try. You might like it.