If she'd been asking the organisations that Alonza actually belonged to, MAthilde probably wouldn't have gotten much, but she didn't. She asked the Colleges and the Dwarfs, neither of whom are Inquisition organisations.
Not really? Like, I imagine that if the Colleges get someone going "I'm in x, there's a Witch Hunter poking around, what do we know about them?" they're perfectly happy to respond. They'd probably send someone out there to double check the Wizard in question isn't hiding soemthing, but the most common response if you are is probably to kill the Hunter and hope no one finds out (which is a solid strategy tbh) not ask the Collegs what their track record is.
I did say I thought that Gretel would have gotten the info much slower. And she might not have been able to draw the right conclusions, but that's all about personal skill, not organisational connections.
Not really? There's an enourmous difference between attacking one city and permanently occupying large parts of the Empire. Also, have they? I know they beseiged it once (well, Arkhan did) but he didn't actually sack the city, he just used the seige as a distraction to break in and steal a book from the vaults of the Cult of Sigmar.