If anyone has experience with any of these three models, I would love to hear your opinions/experiences. I am trying to decide which one would be the best purchase. I've read customer reviews that complain about noise--is it really as bad as they say?

Perhaps the main problem compact cameras experience is digital noise, which is made worse by companies selling you higher resolution cameras (more pixels) on smaller sensors and without engineering the other factors (processor, NR, etc.). Check dpreview.com and specifically the review for the Canon G9 (it mentions other compacts offering great IQ, Fujifilm in particular, but I've forgotten which model). If you're not planning on printing A3, an 8 MP good IQ compact is better than a 12 MP poor one (and cheaper).
Other pointers:
The Lumix models may have a Leica logo but they are only 'built to Leica specs' - don't be conned by that nane...
Compacts are terribly slow to focus and need quite good lighting levels - a problem if you're shooting children and/or indoors.
Compact lenses can suffer from radical chromatic aberration (bizarre colours along edges) and distortion, so you'll need good PP software to try and offset the problem.
If size isn't a problem, I really would recommend a budget Nikon or Canon dSLR costing only a little more which you can use in automatic mode and then perhaps improve with later.
If you've really narrowed it down to the three you mention in your title, all I can suggest is to go with your intuition - which one feels and handles right? They all use basically the same sensor, so everything else is subjective in the end.
Hope this helps.

I didn't think there were dSLRs in the $300-$400 range, which is the maximum of my budget for a new camera.

I understand that Nikon's low-end SLR and the Olympus e-400/410, etc., might be in that range. It may be a good idea to find an "old", i.e., last year's, product, such as the Canon 400d kit, which is being replaced by the 450. Adorama falls in your price range. (This is not an endorsement, by the way.)
Olympus uses the four-thirds system, so the sensor suffers from a 2x crop factor (forget macros). Remember that slrs require more input for more output, but that's a no-brainer compared to the noise on a compact. I just purchased a G9, a flagship compact from Canon for when I can't take an slr with me. Anything above ISO 200 and I'm wasting my time.