No it shouldn't be across three pins of the transistor.
No, there is some fault.
The green dots are an indication that it is connected. Big black dots in SCH are junctions, but be sure to grab the dot and move it to see if traces are actually connected.
NO, one pin of a part should not light up both 12V+, 12V-, and GND.
I don't know how VAN did that .fzz, because it seams to be a modified original.
If you want to upload VAN's .fzz that you are playing with I can try and see what's wrong.
Here is a big .fzz that you can play with to see how it's supposed to work.
https://www.mediafire.com/?oo9d8jbcjpmplag