How to run a (provably probably) secure global referendum?
Regarding the 3rd part of your question, "Can and should ordinary, run of the mill elections be conducted online ..." - If one thinks of an election as a problem around the issue of preserving the privacy of many inputs (a people or population's votes) while correctly producing the right result (i.e. correctly tabulating the outcome of the election) and moreover, if you would like to do this without relying on a trusted central authority (like the election commission) - there are some central results in cryptography that prove the possibility of such a thing under different adversarial assumptions, starting with the landmark paper of Goldreich, Micali and Wigderson entitled "How to Play Any Mental Game - A Completeness Theorem for Protocols with Honest Majority" - this result falls under the general moniker of "Multi Party Computation" (MPC), and represents a provably positive result for MPC with honest majorities. 
Since this paper, MPC has evolved quite a bit and has been promoted, among other things, as a means for secure voting - the more recent SPDZ protocol of Damgard, Pastro, Smart and Zakarias is secure against "active static adversaries" and is able to tolerate corruption of up to n − 1 of the n parties involved in the computation. 
In terms of successful applications, the following outlines one of the first applications of MPC techniques (the Danish sugar beet auction) "https://www.cs.purdue.edu/homes/aliaga/cs197-10/papers/bogetoft.pdf"