I just came across this and had some insight to this conversion. Though I dont know if its been done to a MKIII before, it has been done to a MKII a few years back. I had come up w/ this idea back when the MR-S came out; reading a magazine article on it, they pointed out that it used electric/hydro power steering which I thought was a great idea. After looking into it, I found you could get a '90+ complete pump for about $50. I never got around to doing it on my own car but discussed the idea w/ a friend who wanted to try it out on his MKII. We had already done a 7M-GTE swap and didnt have the power steering hooked up anyway.
We got the pump, mounted it in the area under the D/S headlight (in front of the wheel well), had new hi-pressure hoses made up and wired it up. Its only set up to be on or off, but had discussed having it be on an adjustable switch so that the response could be dialed in. BTW, he was just using the the stock 7M alt. and never had any issues.
Also, you questioned how loud its going to be... This one was just mounted right to the body of the car and it could be heard when running (nothing extreme - like a walbro); but I feel that much of that is due to the pump being mounted right to the sheet metal. If some rubber isolators were used, I think it would make it as quiet as it is in the MR2.
Also, some further food for thought to improve on this idea more. The SW20 pump is pretty heavy as its cast (like the stock 7M pump), but the one from the MR-S is aluminum. I've did spend a little bit of time looking into this and from what I understand, they are both similar set ups, just that the MR-S set up is lighter.
I think the ultimate way to improve the power steering is w/ trying to go w/ a true electric power steering set up (they are usually incorporated right into the steering column or the rack itself). This is what most of the newer cars are coming out w/ and it has numerous advantages - simplicity, weight, no parasitic power loss, and ability to control/ adjust amount of assist. I came across a write up from this guy in the UK that has an early '80s lotus who put an audi V-8 and 6spd in it (did this in his driveway - very impressive). Recently he has swapped in the electric column from a newer GM and it seems to have worked out amazingly.
Good luck w/ it.