Things to do (according to my order of importance):
1) you are describing very well your problem (maybe that is due to the fact that your are a native English speaker), so why not contributing to Acrobat improvement here:
https://www.adobe.com/cfusion/mmform/index.cfm?name=wishform
I myself posted a few suggestions and possible bug reports. They never give a feedback as advised in advance (engineers...furious and wild people...);
2) I myself am not a network expert but you need some help here at this point; find someone skilled in Symantec Endpoint Protection and ask for assistance;
3) "If I try to log in with a different account": where do you log in?
4) "If I use Outlook (for a different email address), a draft message ends up in the Outbox, but I can't send it.": this is really strange and probably not related to Acrobat. Does your document have any scripts or attachment embedded? Do you get any error msg?
How do you have Outlook to open when clicking the envelope icon? Are you manually switching email application under your system settings?
5) "At this point I'll have to have my reviewers download Adobe Reader XI so that they can make comments.": how else would they make comments otherwise?
6) there's probably this workaround to all this; see this tutorial: http://idea.uwosh.edu/nick/distributed%20forms.pdf
now watch the part where they place a button on the document to email the document. You may try to place 2 buttons, one for each account. If this way you get what you need, there are even better way to do the same, that is placing those 2 buttons on the tool bar. In other words, you don't need to keep those buttons on the document but you can have them as extra buttons of your tool bar. However, this last step requires some scripting you may ask someone of this forum to write for you.
Let us know.