dragon!alt-Medea (which her normal form couldn't summon I think?)
Yep. A gift from her patron. She also gained the ability to summon the dracoliches she used against the city/the military, all sort of lesser "mook" versions of the gold dragon.
I'm not sure I understood all of Artoria's deductions though. The melon soda I get, and the explosives, but the part about Assassin's blade and Rider's wounds kind of went over my head.
Thanks to her battlefield experience, Artoria is familiar with the various kinds of wounds inflicted by different weapons. Unlike Tom, she was able to tell that the wounds on Rider and Shinji's bodies were created by a short-handled axe, i.e. Alt-Assassin's hatchets. (Perhaps due to the relative width of the cuts or by signs of bone fracturing, since axes tend to inflict more blunt force trauma than swords do.) She could contrast this with the precise, relatively narrow cuts one might expect from Assassin's
nodachi, given his combat style. Finding said
nodachi embedded in Shirou's head, then, made him the odd case out -- the only instance of that sort of weapon being used at that particular scene
[*]. (Since Alt-Assassin doesn't have access to it.)
Later on, when she saw Archer take out the island wards, she realized that he could (a) replicate swords and (b) fire them from his bow at long range. Which meant that he could have both replicated Assassin's
nodachi -- since he'd seen it briefly in passing before bailing out Tom in Part III.1 -- and sniped Shirou with it from his position on top of Rin's hospital. Add in the other two murder attempts she already knew about, and, well...
Does that make sense? Sorry that I wasn't able to make all of this clearer in the actual story. I was worried about that scene being too slow as is, so I may have rushed through things a little too fast. Hope this helps.
[*] Though I didn't emphasize this at the time, she also thought it was weird they found the blade driven into Shirou point-first, since Japanese swordsmanship is all about slashing. That clued her into the fact that the nodachi was being used in an unusual way. And indeed, while Archer of course knows how a katana or like instrument is supposed to be used, he couldn't replicate the proper slashing motion while firing the sword from his bow.