How do non-MFT lenses work on a Panasonic GF3?

I have a G3, and have really enjoyed using my 3 Pentax K mount lenses with it. Apart from the physical adapter, you will also need to tell the camera to "Shoot without lens" in the REC menu. Apart from the cost savings, you will also get benefits from those old lenses. They are mechanically much more solid, and also (at least for primes) even if you give you more light (lower f stop) to be more successful in many of your shots. You can take any type of shots with manual lenses - it isn't as if sports photography started in the 1970's (when AF lenses started to become available). For those situations you might set a focus point and wait for the athlete to intersect with it, or use a smaller aperture taking advantage of bigger depth of field.

Mirrorless cameras are very good with manual lenses, because they always have a live view (unlikely DSLRs with optical viewfinders) and at least in my G3 provides a very large (10x) magnification to get you focus spot on.

I have a few of my shots with "legacy" lenses here.

http://www.flickr.com/photos/27357592@N04/sets/72157628758857049/

Legacy lenses tend to be much larger and heavier than native m43 equivalents. The primary (and perhaps only) benefit of legacy lenses is the low price. Inexpensive primes from old 35mm cameras can be fun and cheap, but new lenses almost always have better ergonomics and image quality. Avoid old medium-telephoto zooms as they tend to be huge, while native m43 telephoto zooms are tiny and quite affordable.

I have a GF3 with a micro 4/3 -> Pen-F adapter and a 1:1.2 44mm Olympus H Zuiko Lens I took off an old Olympus Pen-F camera. Works great in manual mode for photos. For video I have been unable to activate deshake so handheld videos are unusable. Which is really annoying as deshake is active for the preview when I'm not filming...

---

Remember that when using non-MFT lenses on a GF3 (I use old Pentax-K ones) you lose stabilisation, and both the f-stop (older lenses with aperture ring are best) and focus need to be set manually on the lens. As mentioned, "Shoot without lens" needs to be set, and use Aperture priority (or Manual) mode. Mine works well with a sharp 50mm f:1.4 and a FOTGA PK-M4/3 adapter (though its much heavier than my Panasonic lenses).

Are there any problems with attaching non-MFT lenses?

No, though you will not have access to any electronic/auto features of the lens such as AF, stabilization etc, you'll have to do everything manually. You need to take into account the crop factor, lenses for smaller systems like 8mm, CCTV etc may vignette or produce a circular image with black borders and the focal length will be longer on m43 body.

How can I see the image from the lens without a viewfinder? (Is it going to show correctly on the display?)

Yes, just like with m43 lenses... the light goes through the lens onto the sensor and the sensor image shows on the LCD display.

Can external lenses be used with Raynox DCR 250?

Do you mean using a non-m43 lens with the Raynox DCR 250 and attaching to m43 body? I think that ought to work, the Raynox is macro-adapting the lens... as long as the lens can be fitted to your m43 body then you can see the result of that.