I am certainly not an expert on this but I do work with medical data and also with companies that at times have very stringent data protection requirements, like working in a kind of Faraday cage, of course without internet. Laptops have to be reinstalled, then you work and then you delete everything again.
I found that I can work very well with the offline features of Mathematica. I also have Wolfram Desktop, but only the online-file-version that comes with the Developer package, i.e. I cannot locally safe data, which might be a problem in your case. I can, however, work with Wolfram Desktop and then Select All, Copy, Paste into Mathematica 9.
Regarding your questions:
1) If it is never connected to the Internet, what functionality in Wolfram Desktop would be missing?
Well, some functionality would be lost. You could not make any API requests to Wolfram Alpha. You could also not use Cloud Deploy - which you obviously do not want to use. If you have the Developer Plan you would only be allowed to save files online. There is this "dirty" workaround I mentioned above but that is not really clean. Anyway, most of the features you could wish for when analysing medical data will work perfectly fine offline. Also when you start using Wolfram Desktop you need to "login" with the internet on. Then you switch it off and Wolfram Desktop keeps working.
2) Will the Wolfram Language include the ability to interact with Microsoft .net Framework via NetLink?
No idea. I do not use that.
3) How will users of ExcelLink be affected?
Well, I can certainly import Excel files with Wolfram Desktop. Also features like Dataset work just fine. If you want to access a database it would need to be in a protected local network I guess.
I think that Wolfram Desktop would meet your requirements if you take the Producer level or higher. Alternatively, you could use Mathematica 10 when it will be released - hopefully soon.
I very much agree with you that
"I noticed new language features such as DataSet. This could be extremely useful in current projects. Can these new features be used in Mathematica?"
I think that you should be able to use the features you need for medical data analysis offline with Wolfram Desktop. I think, however, that you might want to rather use Mathematica 10, when it comes out - hopefully soon. I think that that is the real offline product. For your analyses the online features will mostly not be required, I guess.
You might also be interested in this thread.
I think that you also might want to contact your local Wolfram Sales Representative. They can put you in contact with technical staff who can help with all details about how to keep data safe and comply with confidentiality clauses etc.