The iPod Nano can be knocked around pretty good and still keep going, but getting them wet is another story. The units have moistures sensors (much like nearly all of Apple's handheld line) are not to be got wet. The phone strap would provide some protection against this, but it doesn't make it water resistant or turn the device into an actual wristwatch capable of getting weathered. Washing your hands with the phone attached to your wrist would certainly get water in the unit (potentially leading to damage and inoperability) or at the very least, trigger the moisture sensors.
An iPod Nano is still an electronic device that should be kept clear of all water and as much perspiration as possible. If the moisture sensor is triggered, your warranty is void. With that said, the units are sealed quite well and would typically only allow water in the headphone jack and the proprietary cable connection port at the bottom (these are were the moistures sensors are located).
Most tiny radios use the headset as an antenna, and as Apple points out explicitly, so do they.
Regular speakers and other earphones would work similarly, though perhaps not as well as the pair that came with the nano, as it likely has a specific length to best pick up FM transmissions.
All wires receive radio signals, and depending on the configuration and length they are better or worse than other wires. In the case of the nano and similar FM radios that use the headphones as antennas, they use filters to couple the captured RF from the headphones into the FM receiver. The filters usually consist of inductors and capacitors and permit only the RF energy to enter the FM receiver, while blocking the speaker output from the audio driver.
The audio driver may have similar filters that block out the FM energy from the audio driver. The speakers are natural filters - they won't react to signals much above 20kHz, and the radio RF is too small to make a difference even if they did perform at 95MHz.
The cable company does the same thing when they send digital TV, analog TV, phone service, and internet to the user, and the user sends back pay per view requests, and internet and voice data. They all use different bands of the total cable bandwidth and they are separated by filters on each end so the right data gets to the right spot, even though they all share one wire. If you have DSL internet service and take the filters off your phones, you'll notice additional noise on the line - because you are now letting the phone receive the DSL signal that was blocked by the filter.
Best Answer
Nanos or shuffles are great for workouts at the gym or while running. You can use them just about anywhere. I personally use a Classic for my stuff, but the nano is great if you have a smaller library or want something very mobile.
As far as what you can do with it (other than playing music), it really depends on which generation you have. You obviously can't really run 'apps' on it, but many generations (of nanos) had some small games, radios, could play video, etc.