Hello there and welcome to the community 
And lets start right off the bad news. Hunter is in a pretty poor state now. As a class it can offer only single-target damage and CC, skills overlap with their utility, pet AI is still not good and bugs are pretty annoying. Devs are stubborn in their unwillingness to give it a proper rework so community hype over it is now somewhere between zero…
…and zero.
Did that discourage you? If not, welcome to the club. I myself like the idea of pet class very much and Hunter is in my list, although right now
is the ‘best’ buld for Hunter I’m able to come up with 
The point is to take a solid powerful (and independent) DPS class right off the bat and take Hunter or Falconer later if they ever become of any use. You still will be welcomed in parties and may have fun with double pets and your personal bug army (in all senses, lol).
Now to the questions
- Yes. Skills do not overlap though.
- Only Rush dog. Also falcon doesn’t have any auto-attack.
- Yes, any archer class is able to equip a pistol at lv180.
- It will be stunned for ~5 sec, then it may return to combat but it’s HP won’t regenerate instantly.
- It’s totally outdated.
- You can’t. And even if you could, you can’t use Hunter skills while mounting and SR skills while on foot.