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I've tried shooting without a flash, but no luck. When I set the ISO to 1600, and go shutter priority to get Shutter speed of 1/250 my shots are 2-3 stops under exposed
The only way I can shoot action is with a flash.

Shutter priority is very treacherous, which is why it is rarely used. I really don't understand why it is even supported. I guess it looks like a "feature" so it helps sell cameras. Most digital lenses when zoomed, only give you a stop or two elbow room on shutter priority, so if you dare to use it, CONSTANTLY monitor that you are within the aperture range of your lens. Check your monitor on every shot where conditions might have changed even slightly. NEVER relax and trust it unless you have a very fast lens with a great range of apertures and are shooting under very good lighting conditions.

With aperture priority, when you have your lens at the widest aperture, the shutter speed will always be at the maximum speed that will work in those shooting conditions. Aperture priority is a good choice for all-around shooting, and vital for shooting action.

If you are shooting RAW, you can do the equivalent of "pushing" film in the darkroom, by setting your Exposure Compensation to -1.0EV for an effective ISO3200 to -2.0EV for an effective ISO6400. This will result in controlled under-exposure which you compensate for in your RAW processing program. If you can only get 1/60th of a second at ISO1600, it will pop up to 1/250th at -2.0EV.

Like film, it comes with a cost in the potential loss of shadow detail and certainly higher noise levels. I would always prefer a sharp shot with loads of noise over a silky blur. You will need to test under the actual shooting conditions to determine the limits where it gets too ugly for you. Come to think of it, if you were shooting RAW, the underexposed shots you mentioned above might be salvageable.

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I want to take my "old fashion" hand held "Gossen Luna Pro" light meter to the gym and see what kind of setting I should be getting at ISO 1600.

You have an even better light meter built into your camera, with the histogram display. Using test shots and the histogram, you can achieve a far more accurate exposure than any hand-held meter ever could. I believe gentle Ansel would have killed for a histogram!

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I've resorted to using the FL-36 flash which gets me a shutter speed of 1/160, but that, in combination with the slower auto-focus, does not get me the desired results.
I'm very frustrated with this camera. Had I known it could only be used for "nice snapshot photos of the family -non action" I should have gone with Nikon or Canon.

Prior to answering this post, I reviewed the lineup of lenses for your camera and discovered some a couple of significant items. Olympus has an f-2.0 35mm->100mm zoom, a whole stop faster than any zoom I know of for Canon or Nikon! There is also a 50mm (equivalent to 100mm on 35mm) f-2.0. When shooting basketball, my weapon of choice was an f-1.8 105mm lens - almost identical once the crop factor is taken into consideration. Either lens should get you into pay-dirt.

With Nikon's 1.5x crop factor, there is the 85mm f-1.4 which would be very nice, but it is a film lens, and I have no idea if it would work well with a digital body. Also at f-1.4 focus would be critical, so you would either need to go to full manual, or may need one of the two high-end bodies - now we are talkin' some real money here.

Given that there are f-2.0 lenses in appropriate focal lengths, you have access to the most critical element, and perhaps an edge over Nikon and Canon shooters.

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Had I known it could only be used for "nice snapshot photos of the family -non action"

Other than a 1950's box Brownie, that simply does not apply. Even cheap point-and-shoots are way beyond that. Loan me your camera for a week, and the pictures taken will leave your jaw on the ground.

My main camera now is a Nikon Coolpix 8400, a non-dSLR, of a few years vintage, in roughly the price class of yours. Take a glance at my web site and see what our cameras are capable of doing.
CP8400 Photography
Las Vegas Interiors and Street Shots With the CP8400
A Variety of Shots, Mostly With Coolpix Cameras, with notes and links.

I expect that with your camera in hand, I could pretty much duplicate every shot on these pages. They go WAY beyond "nice snapshot photos of the family -non action", and this is a Nikon user talking. (I do have a lovely little Olympus PenFT in my collection.) Feel free to explore the site, there are loads of tutorials, a few essays and more portfolios for ideas and hopefully, inspiration.

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Now for the next question?
I notice the other newspaper photographers covering high school events use mounted flashes that fire (and re-cycle) at the "burst" speed. Are there any other manufacturer's flashes that can be used with this camera.
Using regular Alkalyne batteries I have to wait 15-20 seconds for a recharge.
Any suggestions of flash and external battery pack combo's would be appreciated.

Again looking at Oly equipment, there are three external flashes that probably communicate with your camera. Not really sure which would work best, I will leave that to you to research. Built-in flash, on any digital camera is necessarily limited.

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I'm going to suffer with this thing (E-500) until I can upgrade to a different camera.

Make sure that you thoroughly learn it before making a change. You are using it under extremely challenging conditions - something I did for more than a dozen years. The difference in cameras is much less a factor than differences in skill levels. When I walked in the door to shoot my last game, I was still refining the craft. There is no magic sports camera that will pump the necessary light into a dark room. It is a matter of wide aperture lenses, high ISO speeds and the skill to extract pictures reliably out of impossible conditions. It only comes with time and practice - trade for another camera, and in many ways you will just be moving back near the beginning of the learning curve again.

Joe, you might like to try the rechargeable AA batteries, if have not done so already. I use them in my FL 50; they seem to last longer and recycle faster.
I'm a wedding, not a sports photographer, but I remember reading an article several years ago where the pro sports photogs set up multiple flashes around the arena and trigger them with a radio remote.
Nothing beats access to both the venue and the subject.

They are strobing the arena with professional studio strobes mounted on the ceiling, not on camera type of flashes.

That's done all the time now if you look at any of the current basketball magazines on the news stands. It's the reason I stopped shooting sports in 1978. And the reason my mid 1970s sports pictures now sell so well.

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I remember reading an article several years ago where the pro sports photogs set up multiple flashes around the arena and trigger them with a radio remote.

Just fyi.... If you set your FL 36 flash to Super FP mode, you can crank up your shutter speed past 1/160. I was just reading about and playing with this today. Not sure how well it will all synchronize. I haven't tried during a BB game yet, but will probably try. I've had some decent, no flash, photo's with my 50mm F2. With ISO at 1000, I can get shutter speeds of about 1/340. But as mentioned before, the E500 seems to be pretty noisy (even with noise reduction on).