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As I mentioned in a comment, a relatively new application of multi-party computation is its use as a countermeasure against (mainly hardware) side-channel attacks. In particular, there is a method called "threshold implementations" which is based on a form of secret sharing. A relevant reference would be the paper Secure Hardware Implementations of ...

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Yes, preprocessing Beaver triples in an offline phase leads to a faster online phase. The online phase of an AND gate requires just two openings plus local computations. But there are other advantages as well. Define a "linear representation" $[x]$ to be any way of representing/distributing a value $x$ among parties such that the following properties hold: ...

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Sure. Parties: ​ more than 1 Input: ​ each party has a secret input Output: ​ empty string "Protocol": ​ Each party broadcasts their secret input then outputs the empty string. That is robust for arbitrary thresholds, but is not private against even a single semi-honest adversary.

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