Is the "do" emphasized here?

Is "it's" skipped before "easy for ..."? Can I say "It's easy for you to say"?

Yes, "it's" is omitted here. You can say "It's easy for you to say".

Is the "do" emphasized here?

Yes, in this video the word "do" is emphasized a little here. Aladdin is amazed that magic carpets are real since he has never seen one before and probably never believed in them.

What do you mean by narrowed it down?

'narrow down' means to reduce the number of possibilities or options. It is usually used in the form of narrow something down / narrow down something. Something can be replaced with possibilities, lists, options, candidates.

Is it okay to say "fellas" or "you all" instead of "family?

"Family" in this context is used because he is addressing multiple members of his family here. It would sound a little awkward to say "fellas" because "fellas" is only really used when addressing multiple males and here he is also talking with a woman. "Fellas" also sounds a little old-fashioned. You could replace "family" with "you all", although it would again sound a little formal. So it would be better to replace "family" here with "you guys" or "everyone". Ex: Hello everyone! Ex: Hello you guys, how are you all doing?

What does "stake" mean when it's used as a verb?

"Stake a claim" is the full expression, and this means to assert or declare one's rights or possession over something. The speaker has used this expression to express in a metaphorical way that fireweed is the first plant to arrive, and is earlier than other plants of flowers. This is why it is called "summer's timekeeper." You can use "stake a claim" in similar contexts, like when you want to emphasize that something or someone was the first to arrive or declare its authority over something. Ex: European powers began to stake their claim over North America in the 17th century. Ex: My dog always pees at the fire hydrant in front of our house. He's staking a claim over his territory.