It's possible that they will charge you for a full replacement phone after they get it back if they test the phone and find it rooted. That specifically voids your warranty. Also, did you tell them you dropped it? They could also charge you for that, since it's not a warranty issue, if they decide it's not a warranty issue but an abuse issue.
I don't know if they do any forensics when they get the things back.
I'll be impressed if they say that the screen not working after a drop it is a warranty issue. For the most part, I've been happy - and even impressed - with Verizon's CS lately. A far cry from 10 years ago when I first subscribed!
The refurb I received was perfect (as were all 3 that I received), but it still didn't do wifi, so after one more D1 refurb (the best of the refurbs... except it didn't do wifi), I asked for a different device. Now I'm enjoying wifi without issue on a refurb Droid X, though I miss a lot of the things that you can do with the D1 but not the DX (sync FB contacts with only phone contacts (rather than importing ALL FB contacts) for example).
But I digress. Each of the refurbs I received were perfect (again, except the wifi issue that prompted the replacement in the first place). The last one looked brand new. I put the first mark on the back of the slider when I opened the keyboard. The X I finally got is a refurb and it's also perfect.
I wouldn't worry about refurbs. From watching this thread, it sounds like there may be some small percentage of not "like new" that make it through the process, but they are getting better as the refurb process matures (we tend to complain about bad stuff but forget to say good things when things go as they are supposed to!).