Great questions Nick!
And yes hiding your address, per the
Google Guidelines is MANDATORY! In my opinion hundreds of thousands of listings have been deleted. Many have been down for months. AND even businesses that tried to start over with a new listing in a new account have not been able to get a listing back up. So this is VERY serious business.
When in doubt HIDE YOUR ASS (address) is my advice!
Now on to answering some of your Qs.
Q - Is the whole address supposed to be dissapearing?
A - Yes, the whole street address but city, state will still show.
Added point - Also the map marker will move to the center of the zip or area you select in the service radius selection. And it will change shape and be a circle instead of teardrop.
IMPORTANT POINT - After you hide address, it can take about a week or so for the live listing to update to reflect the change
UBER IMPORTANT POINT - Even after you hide address and it updates, your client COULD still get a moderation call and be deleted. I believe the moderators in India are working off outdated lists and NOT the live maps listing. So still need to coach them that if they get a call, need to be short, direct and emphatic. "No I don't see customers here BUT I already updated my listing to HIDE my address. My listing meets the guidelines, please do not delete it!"
Q - Does this effect the the citations you have for the address?
A - My suspicion is that this does not affect citations. Google still knows the address INSIDE the dashboard, even if it's hidden on live listing. (Similar to the fact that you can still rank #1 for categories in your dashboard even though Google stopped showing those on the live Page too.)
Q - I understand that it's necessary to do, but how does it change the other factors? And how does it change the citations you have already built, the address on the website etc?
A - The main thing that changes is the display, both on the listing and on maps. I have not seen a big change in ranking. In fact I've shared an example where the top 3 A, B, C in one market are all hidden and the lower listings all have addresses displayed. (Cant remember the query but can find it if anyone wants me to.)
I think you should still build full citations and have full NAP on site that matches what's in dash.
The only question I get about this, which you have not asked but other's have, that I don't have a good answer for.
ADDED Q - What if I have a client who works from home and does not want their address displayed on any citation sites or anywhere on the web? Most of the citation sites require an address. How do I build citations for a G+ L page with hidden address if home address is in dash but they don't want home address in citations?
A - I don't know.
Any other hide your ass* (address) Qs I can answer? 
(I think I have answers for most Qs except for that last one.)
Any other thoughts, tips or experiences to share???