Jean D. Morency, DMD

Keeping in mind that the photo only gives a limited amount of information, there is no other GOOD alternatives.  Veneers and or crowns would at best be a compromise in esthetics and function. Dentures would really be a poor choice.  
my question to you is why don't you want braces?

It could be a dry socket.  But it could also be pain from the trauma of the surgery.  Other signs of a dry socket.
is the pain radiating to the ear?  Is the blood clot still in the hole or does the bone look bare.?  A dry socket often has a very specific smell.
I would consult your dentist.

Yes.  Y you are eligible for Invisalign.  I would not recommend Invisalign Express for your particular case.  Invisalign full would be recommended.
very difficult to correct an overbite with Invisalign

It will be hard to give you any advice without examining you and without an X-ray.
Yes, you can have root resorption after you have a root canal. A root canal only remove the nerve inside the tooth.  Resorption eat away the tooth itself.
If you don't feel confident with advice you already received, get a second opinion.

You need to see an orthodontist.  Your "snaggle" tooth is nothing but a tooth pushed out of alignment because the whole front section of your bottom jaw is overcrowded.