Can braces or invisalign without extraction work for me? (photos)

You can definitely do braces or Invisalign without extractions but I would definitely recommend arch development BEFORE braces or Invisalign. With arch development you would eliminate the near posterior crossbite and the over splayed anterior teeth.Good luck!

You have protrusion, crowding, an open bite, and excess overjet.  Braces will align your teeth whether or not you get teeth removed. 
With out extractions, your teeth will stick out further in front of your lips, the front teeth will slant further out,  your open bite will be worse, and if you lose or stop wearing your retainers, your teeth will begin getting crooked again very quickly., but your teeth will be straight!  
If you have braces with extractions, you have a good chance to end up with an ideal bite, i.e. protrusion reduced, ideal overjet, open bite significantly reduced or closed completely (open bites are hard) as well as reasonable stability.  Also, you teeth will be straight.
Choose which ever option meets your overall desires.

I always feel it is best to treat as conservatively as possible.  Extracting teeth for orthodontic therapy is pretty aggressive in most cases, and I know your case can be done without extractions.  I could straighten your teeth without extractions and address the concerns you have while giving you an improved and proper functioning smile.  The only thing that may remain by not doing extractions is some overjet. The concern you have about a flattened face will not be a concern if you do not have extractions done. If you do not want extractions, you should not have then and you should find a provider who can treat you without planning for them.