On a meta-level, I think the story that people like to tell is that Mozilla chose increasing executive compensation rather than using the same money to keep the Servo (and people working on Rust itself) employed.
[0]: https://servo.org
1. I don't know if Rust is being used in Firefox today (edit: turns out it is, see my sibling comment), but I can imagine why they wouldn't: Mixing C++ and Rust is probably non-trivial with the kind of portability they're aiming for. A lot of the UI and "business" logic in Firefox is, and has pretty much always been, implemented in JS and XUL (which was also available to extensions before WebExtensions, which is why Firefox extensions could back then change almost anything about the browser). So with JS available for a lot of the logic, Rust is probably mainly interesting for replacing the lower level C++ parts.
2.Back when Rust was created, it was used in a project called Servo - a new browser engine. To my knowledge there were never concrete plans to switch Firefox over to that engine, but Firefox adopted a lot of the techniques they pioneered. Mozilla was a different organisation back then, happy to experiment with no clear path to revenue. I didn't find any of those developments too surprising. I'm definitely not surprised they don't undertake a major incremental rewrite these days where it's more about fast profitability.
You might be thinking of Servo, which is a (purely?) Rust rendering engine that Mozilla abandoned, though it's not completely dead.
Narrator: It wasn’t the original use case.