Door will not close - latch on door seems to be flipped around so that it bounces off the corresponding piece on the frame rather than catching it to keep the door closed.

Did you look at door latch on door? In the old days sometimes, a piece for the door latch assembly would get turned down, when that happened, the door would bounce off the door stud on door frame. Sometimes you could take a screwdriver and flip that arm up, then the door will shut. The door stud on the door frame has to go into a slot on the door latch mechanism.
Granted, I can't see anything from here? I may be barking up the wrong tree? Just a thought.

Make sure your alarm isn't armed. In many car door latches, there is a metal piece the 'closes' around the pin. If 'piece' can be flipped to the closed position with the door open, and then when you slam the door, it does not latch.

Open the door & have someone hold it open,
with the door handle pulled out

Use a screw driver or any round tool to open the
plastic lock catch in the door latch

Sometimes your clothes catch it & close it ,as if it
were wrapped around the B-Pillar Pin, your referring to

Door pins are a common failure on that series of Blazer/S10's. Also, another point of contention that the door faces is a sticky latch. Basically, the door latch re-latches itself and when you shut the door into the catch, it slams into it and bounces off. An easy fix is a 3 part cocktail of brake parts cleaner. Open and shut the latch with a screw driver acting as the catch and your interior door handle a few times. P'Blaster, open and shut the door a few times with the screwdriver acting as the catch. Finally, a good shot of white lithium grease. Open and shut the door with a screwdriver as the catch several times. Once you feel it is un-stuck, actually close and open the door several times. If you notice the door rising every time it catches, you MUST replace the door pins (cheap fix). If the door still has issues, replace the door latch (and since you're in there, do some PM and replace the actuator as well).

with a screwdriver flip it back

there is a spring inside the door lock that makes it work this spring is most likely broken try spraying lubricant on the lock just in case its binding ,but it seems like you will have to replace the lock mechanism

Look on the very end ot the door you will see where the recieving latch is.This is where you spay.You need to make sure that the latch didn't fall into the closed position.You can look at the latches on the other doors to see what they look like in the open position.You can take a flat screwdriver to flip the latch back and forth while you hold the handle.

Look on the end of the door and you will see the slot where the door catch is located. It should line up with a big stud sticking out from the post by the drivers seat. The gap should be open. If you see a piece of steel blocking the slot, then the door catch is already in the latched position. Insert a screwdriver into the bottom of that catch and pull it up into the open position. If that is the problem, the door will work after that. Hope that helps

With the door open hold the door handle up as if opening the door. The latch in the door mechanism will now be free so you need to rotate it to the open position. It might be a bit stiff but it will rotate with the handle pushed You might need a pair of pliers or small wrench to move it around.

Check the latch mechanism on the door. There is a piece of the latch that closes around the clasp on the door frame. sometimes this closes when the door is open and you can't shut the door. Take a screw driver and flip this latch piece open and close the door.