I use a stainless stone and just put it in iodophor to sanitize. I recommend a 2 micron stone, not 0.5 micron. The 2 micron ones are much easier to clean and sanitize because flow through them is easier. I just flush the whole assembly with water after use (I put a little barb on a hose adapter on my sink faucet) and run air or CO2 through to mostly dry it out.
I've used air, and now use the red Bernzomatic oxygen cylinders. The air coming out of my aquarium pump stank from the rubber diaphragm, and I don't want my beer to taste like rubber, so I rigged a carbon filter as well as a sterile filter in line so I'd just get nice, pure air into the wort.
The one handling instruction I know of (never heard of something intended to go in liquid that couldn't get wet) is not to directly touch it, as the oils in your skin will clog the pores -- handle it like a halogen headlight bulb.
I haven't ever tried it, but the idea of using a stirrer on a drill ought to be a lot easier than pouring back and forth or shaking, and a pretty clever, simple idea. Easy to clean and sanitize too.
As kramerog indicates, the maximum dissolved oxygen level at saturation with air is around 8ppm, which is just about enough in most cases. 1-2 minutes of gentle bubbling with pure O2 should get 10ppm easily, which is supposed to be ideal. Also note it is possible to go too far. 10-12ppm is really the most you want. The Yeast book by White and Zainasheff has some tables with results of O2 levels they got using various methods. One more point on using a stone. Ideally, you have a flow rate such that fine bubbles are just reaching the surface, not churning up a boiling head of foam -- gas that's breaking the surface and shooting into the air is obviously not dissolving into the wort.
Very strong beers may need more, but rather that try to get more O2 in upfront, the advice I've always seen is to aerate a second time several hours (up to 12) after pitching, while of course any oxygenation after fermentation starts is usually a big no-no.