Rewatched "A Study in Pink" this afternoon. I didn't get the impression that it's a GAS.
Watson is clearly the intended MC, and Sherlock is the IC, but anything more than that starts to get a little fuzzy, in my opinion.
First, there doesn't appear to be any Changed character. It feels like Watson's throughline is meant to be about his PTSD, but it's half-hearted and pretty thin. He has his psychosomatic limb and hand tremor, but he's given them up or grown out of them well before the final act. Ultimately, his shooting of the Cabbie, which resolves the OS, has no apparent connection to the MC Throughline. Are we meant to believe that Watson's PTSD would have prevented him from pulling the trigger at the beginning? I just don't see enough of a baseline established for us to say for sure. Maybe if the episode had ended with Watson writing in his blog, we'd have an author's proof of his change.
Interestingly, the final lines of dialogue come from Mycroft and concern how Watson might change Sherlock. This points to a larger GAS potentially spanning the season or multiple seasons (not that this means there isn't one in this episode).
I'd guess that Watson is a Be-er, or at least that was the intention. The internalization of his problems have manifested in his limp and tremors. He has nightmares and responds by sitting alone in his room and goes to therapy, which I interpret as an attempt to change himself internally. The therapist wants him to blog, a way to externalize his problem, but he's unable to.
The issue is that this is dropped pretty much the moment Watson meets Sherlock. They go to the apartment, the police show up and invite Sherlock to the new crime scene, and Watson leaps at the chance to go participate. If he's a changed MC, the change happens here. I'm not sure we really see the Be-er Watson again.
Sherlock is a Do-er. He prefers to text, and when his phone is acting up, won't even change himself enough to make a phone call on a landline, preferring instead to borrow a working phone. Generally, he's brilliant but his working relationship with the police is held back by his poor social skills. Instead of changing himself to fit in better and resolve those issues, Sherlock prefers to pile on more and more brilliant observations until he convinces them.
I agree with @Gregolas about the Domains, with the caveat that if it's not a GAS no Domains can really be assigned. It felt a lot like the Silence of the Lambs in the sense that it opened with several crimes having already been committed, with Sherlock as the brilliant psychopath (er, high-functioning sociopath) the authorities turn to for help.
The difference between Lector and Sherlock is that Sherlock's influence doesn't come from developing a psychological profile of his targets, but rather by seeing the physical evidence and drawing conclusions. Lector looks to motivations for behavior while Sherlock evaluates what he sees. Lector is in Psychology, and Sherlock is in Physics. Plus, Sherlock even influences Watson through sheer doing--running through the streets of London all while Watson has left his cane behind.
This puts Watson in Pschology, which makes sense; the limp is a symptom of dysfunctional psychology.
The RS would be in Mind, which I don't quite see. In this episode there is very little conflict in their relationship beyond the running gag of them being misidentified as a gay couple. Watson is quite taken with Sherlock's brilliance and happily follows him all over London.
As for the Problem-Solving Style, it's really hard to say. Watson is given very few opportunities to solve problems on his own. That's not surprising; Sherlock Holmes' whole deal is that he's THE problem solver, and he just sort of sucks up all the air in that area of the story. Overall, I'd say Sherlock is Linear for reasons already outlined in this thread, but I'm not convinced we see enough problem solving outside the necessities of the OS to say definitively. Plus, the firs thing we see Sherlock do is send the text messages to all the press and detectives during the press conference. This strikes me as holistic. He's trying to change the balance of the relationships at play, so he'll get brought in.
As far as I understand, none of this has any bearing on Watson's PSS, and I don't think we have enough to go on to say. He does just shoot the guy at the end, which seems about as linear as something can be, but again that's in the OS.
Overall this feels to me more like the opening act of a GAS, than a complete story in its own right.