Is this tiny screw the entire implant that came out of my mouth? (photo)

what you have fallen out is the healing cap, not the implant, and sometimes if the screw is not tightened enough , it might fall outso the must important thing to do now is to go back and provably they will even be ready to be restored, or at least re set the screw in, you do not want food to get trapped there.

The part that fell out of your mouth is most likely a healing abutment.  It is used to cover the implant to protect it.  They fall out occasionally and just need to be tightened back on.  It is not something to be overly concerned about but make sure you see the dentist soon to get it replaced.

No.When implants are placed, they normally need to stay undisturbed until the body fuses to the implant (osseointegration).  Eventually we want to access the implant, and to make it easier (and control bacteria and debris) the top of the implant is covered with a cover screw or healing collar.  The choice is made at the time of placement and is determined by "primary stability".  This is "how secure" the implant is at the moment of insertion.  Since what you have is called a healing collar, the implant likely had a fair amount of primary stability.All is not lost in your case.  Inform your doctor, but this usually means leave it alone for another month or so and the next appointment will not be to take impressions but to access the implant and place another healing collar or provisional restoration to develop the gum tissues.