1. How come sometimes, the FSDT vehichles show up, and so so the default vehicheles too, they also run into each other, dissapear (and show up), or have no tires?
2. The jet way sometimes does not show upat a gate I choose, run through the plane, or isn't next to the plane at all.
This has been explained many times already: it's LOD, to save fps, all objects have different less complex versions that shows up in the distance.
They should show up normally, when closer. If they don't, check you don't have a too high value for the SmallPartRejectRadius in the FSX.CFG ( this can be tweaked with the Addon Manager as well). Good values should be 1 or 2.
3. The taxiway shows, but then all of a sudden ends and there is just grey taxiway with no lines.
All objects have a loading range. You can control it with the "Anti PopUp" slider in the Adddon Manager, 1.0 it's the default but, if you put it to higher numbers, like 2 or 4, the default loading distances will be multiplied by that number, lessening the pop-up, at the cost of an fps loss, depening on your system.
Also, how does the park-me work 
Explained in the manual.
