How were the culprits "dragged to their death"? After a student is found guilty, the next scene if of them being tied up or trapped in a new room before dying an "interesting" death.

But who put them there? Does the anime ever give any indication of who is tying up the guilty and dragging them to the "death room"?

See this video for example (Ep3): https://youtu.be/um_HQ6XyzLQ?t=17m
The culprit is found guilty and shortly after Monokuma pushes the red button, he is instantly "teleported" to a room to die. How?

I hope this question makes sense. I couldn't figure out how to word it properly.
The anime series itself shows quite clearly that it is Monokuma.

He is only seen actually dragging the culprit himself in the gameplay-looking animation, but there's no reason to doubt that that's exactly what happens, particulary when it is consistent with the argument.

The following images are taken from the same episode you use as an example.