Welcome to the Q&A Forum

Are these real people?  G+ has a facial recognition and will remove stock photos etc.

This has been gone over several times. Below are some great links to help you with that.

http://moz.com/blog/40-important-local-search-questions-answered

http://www.localseoguide.com/seo-for-businesses-with-multiple-locations-in-the-same-city/

http://www.clickz.com/clickz/column/2283736/local-seo-strategy-for-the-multilocation-marketer

James, Are you running FB ads? Do a quick audit and make sure they are not a bunch of junk likes.  I have seen this from competitors trying to lower Edgerank scores so your ads become unseen by others on FB.

Here is a good read http://blog.nakedlime.com/2012/09/fake-facebook-likes-and-the-trouble-they-cause/

Nope No value. Another SMB scam tactic...

No that would be against G+ TOS. You would need to use the rel=publisher tag instead and link to the business G+ page.

It will help drive some traffic though you need to have high quality articles that are informative and provides value to the reader.  The links are no follow but who cares your using it to gain traffic based on the article. Quit trying to game google and build useful content for the internet.

Try building some followers to FB fan page, Get Likes and make sure you have a branded url for it. You should see it pick up after that.

This is an issue there fixing you can keep track of known problems here http://health.moz.com/ if you have futher issues you can also email help@moz.com

Yep this has already been posted. But you can read the post here.
http://moz.com/community/q/google-analytics-keeps-poping-up-that-i-need-to-re-authorize#reply_205099

No this would not boost the DA for your page.  The link itself would help increase the overall DA but not the CTR on the link. So if you thinking about buying traffic through that link don't waste your money it will only spike server traffic. Best thing is to focus on QUALITY everything from your links, social media, and the major one Useful content. There is no overnight trick to boost your DA and it takes time. But creating value for the end user and not google you will see a rise in Conversions and DA.
You can read more in depth here about the factors that go into play with DA
http://www.searchenginejournal.com/care-domain-authority/

Also read the Basics http://moz.com/beginners-guide-to-seo

Usually fixed within a few days if it's a known error.  Email them about this at help@moz.com and they can assist you further.

Yep I agree 99% with you. Bugs are a normal thing and since the new MOZ came out of beta it has been plagued with them.  There are several other great tools out there that do the same thing and maybe better. One I love is Raven Tools

Certainly! Here is a great read on content length. But again Quality will be the biggest factor.

http://www.quicksprout.com/2012/12/20/the-science-behind-long-copy-how-more-content-increases-rankings-and-conversions/

Still around 700+ There is really not a formula as pages with 300 words rank very well.

This is an issue there fixing you can keep track of known problems here http://health.moz.com/

Are you doing any type of PPC? That can cause some issues with there content network ads.