Rust and Servo were basically announced to the world at the same time, as "the Servo project." It was really important that we build Servo at the same time as Rust, to make sure that the language was actually useful at building a real practical thing. The experience of building Servo helped Rust's development immeasurably. Furthermore, it also helped Rust be built in Rust itself. There's a danger when you build a boot strapped compiler: you might make a language that's great for making compilers, but not much else. Servo made sure that wasn't true.
https://bugzilla.mozilla.org/show_bug.cgi?id=1135640 is the overall tracking bug for moving Rust stuff into Firefox. Firefox 45 on Mac and Linux will contain a teeny bit of Rust, and just got turned on for Windows a few days ago. It's an MP4 metadata parser. The URL parser has a patch, but it hasn't landed yet.
The most notable example is the proposal variously known as "virtual structs" or "inheritance", which had a brief prototype implementation in Rust (circa 2013) solely to support Servo's needs (specifically regarding representing the DOM), but which met a wall of opposition from the community. Such a feature is still likely to appear sometime in the future (after all, if Servo needs it, then it's likely that others do too), but the Rust developers are taking their time to explore solutions that integrate the most cleanly with the rest of the language.
That said, we do take their needs into account rather highly, just like any project with a large upstream user.
Just like we wouldn't want Rust to be useful for only compilers, "only compilers and web rendering engines" isn't much better.
I'm not sure, as I'm on the Rust team, and not Servo; maybe someone who works on Servo directly knows better than I what plans are here.
To capitalise on this, and to give Rust a ‘halo’ project that uncovers potential issues early, Mozilla kicked off a project called Servo to research the ways that Rust could be used to produce a browser rendering engine. This has let them explore technologies like multicore rendering of a single page without dealing with the practical considerations (tried to find a nice way of putting it ) that Gecko has scattered throughout its codebase.
After Mozilla had made some progress Samsung became interested and started contributing a reasonable amount of code. I’m not sure if this is still happening, but you can see their interest given their prediliction for 8-core mobile CPUs.
Mozilla currently have no plans to replace Gecko with Servo outright, but they are folding Servo components back into Gecko when they’re portable and have obvious benefits. Gecko’s build chain can now handle Rust and I believe the URL parser is a port of Servo code (though I’m not sure it has landed yet). I think I recall a Mozilla engineer saying that Servo had progressed beyond a research project now, and it is pretty good at rendering reasonably complex sites, but they’ve still got a long way to go before it could be a drop-in replacement for Gecko.