Yes - it can definitely be a messy job! Not only during, but for a while afterwards it's going to drip - especially during warmer weather. Park over cardboard, gravel, or grass for a couple of days if possible. It's not supposed to stain, but I wouldn't test that claim.
Outdoors is a must, unless it's a dedicated facility. The mist will cling to everything. I wear a dust mask at a minimum, and always try to be "upwind". Avoid really windy days too though - they can get rather wasteful and the neighbors won't like you very much.
My equipment is at my in-laws house. Fortunately they have a gravel driveway there. Let's just say that dust isn't a concern...
I'd strongly recommend getting it done professionally. They have the best equipment for the job, and most are pretty thorough. My father in law already owned the equipment for me to use, so I've stuck mostly to doing it myself to save $$. I have paid them a couple of times to do it if I bought a vehicle at a time of year where I didn't want to try doing it outside myself. It's not exactly the most enjoyable job even in nicer weather.
To get under the car, all I have are a couple of fairly standard steel ramps with extensions to help with low ground clearance cars. It's a bit tight crawling under the center portion of the car, but I still fit. My back, however seems to be less impressed with my choice to do so with each passing year
John