1. I've been based in Miami. It isn't difficult to mistake T1 for the full length of the runway as you're making your way down taxiway T. Does QTR have a moving map display in the cockpit?
2. QTR will usually (but not always) depart runway 9. It's not at all uncommon to see a passenger widebody on a long flight request runway 9. It's a much longer taxi, and I've never heard anyone request T1.
3.Regarding the 105 heading, all departures are assigned a heading initially. Perhaps when departing late at night (after maybe 11pm), you'll fly the SID as published. However, it's typical to be vectored to join the SID. The 105 heading was probably given because of a need for separation with the traffic departing 8R, which is the primary departure runway.
I can easily spot the problem with #T1 and /T1