This article on 4/3 Rumors about the future of 4/3rds is two years old now, but everything said is still true. Although no one has officially turned off the lights and locked the door, the original Four Thirds system is defunct, with both big players (and for that matter, smaller ones like Kodak licensee JK Imaging) supporting the mirrorless camera system Micro Four Thirds instead.
Olympus, at least, has expressed interest in making sure Four Thirds lenses work well (with an adaptor). So, looking at your list:
Large rubber grip, decently heavy
This is pretty subjective. Because mirrorless makes the form-factor more flexible, there are basically three types of body design available. First, the compact-camera inspired "PEN" series and similar. Based on your comments, that's probably not what you're looking for. There are also options which look and handle more like a rangefinder. But probably the series that fits your preferences best is the Olympus OM-D range — currently sporting a higher end model, the E-M1, and a lower-range model, the E-M10.
These use an electronic viewfinder (a tiny LCD), but are styled to look and handle like an SLR. (Before you dismiss that out of hand — this technology has much improved, and the E-510 had relatively small and dark optical finder. This might be an improvement.)
Whether the grip and handling is similar enough to be to your liking is something only you can judge (although personally, I'd caution against first impressions; sometimes these things feel wrong in the store but become perfectly comfortable after a few weeks of real use — and sometimes, just the opposite).
You don't mention it, but if a optical viewfinder is an important part of your preferences, you might be happier switching systems — Canon, Nikon, and Pentax continue to crank out entry-level DSLRs and you can probably find something that matches what you like. (And Sony makes a fixed-mirror "SLT" with SLR-like ergonomics.)
Generally, as you go up the product range, and therefore up in price, weight will follow. Your E-510 weighs 535g with battery; the OM-D E-M10 I mentioned is noticeably lighter at 395g; the higher-end E-M1 is closer at 497g.
Controls have physical buttons/switches
So, here... your E-510 is middle-to-poor in this regard, really. Certainly this is something which continues to be important, but in general you'll get more at the higher end and less at the low end. Check the reviews, but, basically, unless the camera is touting a touch-screen UI, most middle-entry cameras will have controls similar to yours, and as you go up, will be even better in this regard. Both of the OM-D cameras I mentioned beat the E-510's pants off, really, with twin-dial controls.
4/3 system allows me to use several lenses I already own
Okay, so, here's kind of the killer. These lenses will work on m4/3rds, but you'll need an adaptor, which runs around $160. If you have a big investment in great glass (and there are some great Four Thirds lenses!), you'll probably want to go this way. However, if it's a lower or middle-of-the-line selection, you might want to consider selling it for what you can get and switching full on, either to m4/3rds if you find the mirrorless lineup to your liking or to another system where DSLRs are still a going concern.
That $200-$500 price range...
This is at the very low end of realistic. The E-510 was $800 when it was announced. Check out DPreview's entry-level mirrorless roundup for 2014 — you'll certainly find some options in your price range, but you're going to compromise heavily on all three of your points (especially if you consider adding that $160 adaptor). You can get what you want — but not necessarily so cheaply.