Rather than plumbing waste out of each pot, I'd probably add a large drip tray under each shelf and drain that to waste.
In terms of supply, in order to keep up with the ideals of SH culture, you're going to need to provide a pulse of water strong enough to basically fill the pot right up to the top and then stop and let it drain. This is a significant volume of water, particularly as the drain holes make for fairly speedy drainage!
However, as long as you realise that occasional extra "love" might be required to keep up the full pot volume flushing for no salt build-up, you can probably get away with the occasional week or more where conditions are somewhat less than ideal (particularly if you monitor the EC of your water and adjust your fert input, probably downward) - to work this out, compare your "normal" S/H waste water EC with your "fancy automated" S/H waste water.
Essentially, to get water in there, you're then going to need a pump-fed supply line. This will be somewhat ugly for in your lounge, but can be camouflaged into a "side table" with some basic carpentry. The pipes will be more of a challenge.
Get a reservoir large enough for your purposes (work out pot volumes, how many times you want it to work between fills, etc).
Get a pump and suitable pipework - I would suggest you get fairly large diameter pipe into which you tap smaller "feed hoses" to each of your pots; I'd use irrigation or fish tank airline fittings.
Get a timer that can run your pump for the right duration - this will likely be seconds. If a seconds timer can't switch the ameperage required by the pump, you'll need a relay. Remember that if you have 3 or more levels, the lower levels will get more water than the upper ones (although you can play around with valves to help mitigate this) - you'll be better off with a pump per level, although this may get expensive. Make sure the pump can handle the volume and head required.
For drainage, I'd suggest you collect all the waste water into another similarly sized container to the supply reservoir (in case of mishaps...) and have a pump in that container that either dumps it into a sink way across the house, or out the window. You can do this manually or use a float valve. With house-sitters, automation is useful and leads to fewer flooded floors.
You could even put in a relay based float switch "interlock" that disconnects the pumps should the water level in the discard reservoir become too high - also hook it up to an annoying buzzer, which has a note next to it explaining why it's buzzing and has a blinky red light of doom. This is particularly useful if you employ forgetful house-sitters who remember about putting water into the system, but forget that water goes somewhere.
Take a look at the way aquaculture people do centralised filtration, and you'll have an idea of how you can get equal flows throughout your system. If you're not sure, I can probably sketch you something quickly. The more valves you have, the more control you have, but beware of too much "back pressure" on your pump.
Also, run the system for a least a couple of weeks to iron out any kinks before you go running off out of town on business.
---------- Post added at 11:15 AM ---------- Previous post was at 11:00 AM ----------
re: some of your other concerns -
So long as the drainage out of the communal sink is fast enough, the pots are more or less isolated - each pot gets it's own clean water supply line, and the waste water doesn't get into other pots. You may have to play around with how wide your waste is - if one wide won't work fast enough, add more! you can also raise the pots up a little out of the "swamp"!.
Because the supply lines are thin, flexible hose, it's easy to take off the pot and re-arrange. I'm in two minds whether it would be better to push the hose into the pot, or figure some way of holding it above the top of the pot. Perhaps a micro-irrigation head fixed to a small stick might be the way to go, ensuring fairly even coverage of the pot (although if your volume flow is high enough, then that becomes academic).
If you have issues with pot(s) getting too full, dial down the flow or add additional drainage holes at the level you want the water to stop at. It'll change the conditions some, but is probably better than the alternative.
Gravity feeding works, but pumps failing generally means less water on your floor than when your gravity fed plumbing goes awry. Make darn sure your fittings are all properly sealed and fastened. I'm a big fan of plastic pipes that have been welded together with adhesive. uPVC piping works quite well, and you can get all sorts of awesome fittings:
Google tells me your water requirement per pot is about 1/2 a litre (
443ml) That means, with 100 pots, you need a minimum volume of 50l, plus the water needed to fill the pipework (which you can work out with pi*r squared*h, a variable I can't calculate as I don't know anything about your proposed plumbing lengths or diameters). Say another 10-20 litres. That's say 70l per watering - divide that by 4.5 for gallons - roughly 19 gallons - which is not an insignificant volume of water to have lying around in your lounge. Multiply that by 3-4 for your preferred endurance, and things get interestingly large!
Let us know if that answers your queries?