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No, that's perfectly fine to do.

With real estate agencies I work with in vacation rentals markets, I've set up automated emails asking for feedback. If the feedback has a positive rating, we automatically send another email asking them to review us where our local listings are.

Personally, I don't like pushing them to one review source (like Google+) because typically people have their own review preferences, and I haven't tested this but giving them options likely means a higher review rate.

You're typing your domain into what browsers?

You need to figure out if you're pulling suggest data from a search engine or suggestions from browser search history?

Try clearing your browsing history and see what happens.

Will the company name and title tags if your doing company + phrases also match up on both? How different is the content? Same topics and format? What's your reasoning behind having two separate sites from the same company competing for same/similar keywords?

I would try to differentiate as much as possible using separate hosting accounts or c-blocks, or take it even further with separate hosts.

I recently had to do this. The client actually didn't tell me they were moving locations and I found out after the fact. I'm not sure there is a whole lot you can do if you're referring to local/universal placement rankings. The biggest impact on those ranking changes is going to come from proximity changes, most importantly physical address in city and proximity of location to centroid of city. For us the move was a couple blocks so it didn't make a difference.

The only thing I can think of is improving for non NAP related ranking factors, ie. brand mentions, fostering real reviews, domain authority/quality of locally relevant and topically relevant inbound links, etc., and be prepared to announce the move, which a press release would help with.

Ah I'd just keep in mind that YEXT's software has backend access with their partnerships with directories whereas Moz Local just submits to a handful of data aggregating directories, so it takes much longer. I've had issues with Moz Local taking a long time to aggregate data to the directories they submit to. And then clients ask why something isn't updated and we have to go in and update it manually.

Also important to consider duplicate listing issues. Yext can directly suppress duplicates with their partners in 24 to 48 hours. I don't think Moz does that yet.