I'm currently interested in replacing the bridge on my Ibanez S420. I don't use the vibrato arm much, if at all, I don't like the over-complicated method of replacing strings, tuning is a hassle, so on and so forth.

Is it possible to replace the bridge entirely? Normally, this would be a hurr-durp kind of question, but I'm concerned because the guitar's body is really shallow. Would doing this be feasible? If so, what do you estimate the cost to be?

The ZR trem is one of the best features of the S series. It has a very simple string change method....... for a locking bridge anyway and great stability. However to answer your question I would think that it should be fine as long as it was mounted properly because all bridges have support fitted in.
Alright. I'm entertaining the thought because I've had a lot of negatives with the instrument. The biggest thing right now is that the intonation is a bitch to get right. I've discovered that the intonation screws were almost all stripped when I tried to fix the intonation (which got waaaaaaaaaay off at some point). Then it turns out one of the string height screws was not only stripped, but also broke halfway down the shaft. Not a lot of good has come from the thing.
Oh that's awful. I mean if it's new that's a factory/builder fault and I'd consider complaining or something because obviously that's not good enough. I've had my S2170 for maybe 2 years and had no problems. It is a bitch to change the intonation but intonation can be horrible to change on most instruments (I have a strat and a schecter hellraiser as well) and I tend to only do it if I'm changing string gauge sets, no need if I just change strings for the same make again.

If it is faulty and it's an age thing, not a factory thing then perhaps a change would be right but I'll warn you now that I know people that have spent in excess of £400 ($650) fitting and buying all the parts for a new bridge because of strengthening issues, getting the bridge itself, further fitting and repairs.... the list goes on........
The cheapest route that I can see is buying a cheap LoPro or older Edge off of ebay and then block it off. You'd still have to use the allen wrench to put the strings in, but since you don't have to lock the the strings up at the nut, and you more-or-less have a hardtail-bridge now, tuning and string changes won't be as much as a hassle.
The screw problems I just noticed a few weeks ago - guitar was fine until then, but I doubt it's a problem with care and whatnot, since I've had it for a couple years now and have made sure to take good care of all the hardware.

I'm taking it in to get set up and hopefully get the damaged screws replaced tomorrow. If I end up deciding to get a new bridge, I'll bring my questions here.