Yes, Arizona's Lunar & Planetary Laboratory is a top program, even though you may or may not think of the University of Arizona when you think of "top programs".
Caltech doesn't have Earth & Planetary Sciences -- it's actually called Geological & Planetary Sciences (GPS). I'm actually in that program right now and doing similar research to what you are interested in (i.e. bodies inside the Solar System). My undergrad thesis (in Canada) was focussed on NEOs specifically.
If you are only interested in detection/dynamics/origins of these objects and nothing else, then sometimes the researchers will be working in Astronomy departments rather than Earth/Geological & Planetary Sciences. However, in schools with big planetary science programs, the researchers might work in the Planetary Science department or be in both! In addition, if you are interested in more than just the "physics" (for example, composition, comparisons with the material that made up the Earth and other planets, etc.) then you should be looking at Planetary Science programs!
The work isn't all coding. I don't think it's any more or less coding than any other data analysis work. I definitely do a lot of coding and I enjoy it! But I've also needed to learn dynamics (in order to write the physics into the code, you must first know the physics!) as well as actual data collection (from telescopes).
Finally, I don't know if universities actually do specific research in preventing/avoiding/coping with asteroid impacts. I think that kind of work is often left to other agencies outside of academia. From my experience talking to many NEO people, the majority of us do work that "advise" other agencies regarding NEO impact risk by doing things like:
1. Detecting NEOs through surveys (to understand what is out there, how many of each size etc.)
2. Characterizing populations of NEOs (so that when we find an NEO, we can categorize/classify it)
3. Characterizing compositions of NEOs (being hit by a rock made of pure iron would be very different than one made of silicates!)
However, as you mention, JPL is a big player in this field. Although JPL isn't a university, you might know that JPL is actually run by Caltech (under contract from NASA). Many graduate students here at Caltech GPS work closely with JPL scientists and some faculty members here have adjunct status at JPL (and vice-versa). It's actually not too rare for students to work with scientists outside of their home institutions (but usually they have a main supervisor at their home school and this works better when the institutes are nearby -- Caltech and JPL are only 30 mins apart).
You could also try to attend the "Division of Planetary Science" (DPS) annual meeting (in the Fall) -- they have entire sessions planned around "Small Bodies", which include things like asteroids, comets, NEOs, KBOs, and so on! Or, maybe you can dig up past abstracts submitted and figure out who is doing interesting stuff and then trace them back to their home institutions.
Anyways, there's definitely a lot of NEO / Solar System research going on at schools. You might be a little bit harder pressed to find academics working on things to directly related to NEO impacts but there are a lot of work to be done in further understanding NEOs and thus indirectly improve our ability to handle NEO impact risk.