As Ryan suggests you still want to FOLLOW rather than giving the bots a dead end as I notice your heading suggests no-follow.

Has the mobile/skype link been set up incorrectly, not using the attribute?  I'd check the source code to see what the link actually is.  It should be something like -

I'm working on a site that had a very poor disavow file in place from a previous agency.  I went through the link profile with a fine tooth comb and added a bunch more domains to the file.  Organic traffic has gone up over 80%.

That is not our only problem and there are other factors to work on but the cleansing is reassuring even though we know the profile is pretty thin.  But again that just goes to show quality always wins over quantity.

I would say there may be a whole host of issues hampering ranking but a dirty profile is a must fix, no matter the immediate outcome.  I explain to clients that I anticipate a short term fall and why to set their expectations, then anything positive is a great bonus.

"Other SEO's have never had a problem with this arrangement, but this SEO claimed what they do is secret and for no one else to see."

Run a mile!!!

I've just had fresh content crawled and indexed that is in this scenario.  Basically we are saying to the visitor "if you really want to know some more boring technical information then expand this, but we don't want to spoil your experience by vomiting all the data at you at once".  Crazy if that is changed.

A reputable business will have a trading address on their site, therefore the actual trading address and listings will differ.  In this case every aspect of NAP will differ.