Crypto - Wire’s Proteus is an independent implementation of Axolotl that was later renamed to Signal Protocol. There’s also an external audit available - info and links from Wire.com/privacy.
As to business model, Wire is VC funded by Iconical.com /Janus Friis (co-founder of Skype) and announced the first paid product in July - wire.com/teams.
It’s a requirement many organizations have - government institutions are a good example. Some might want to not leak any data to the outside world, run it only for internal use, etc.
One of the details we haven't quite settled on yet. Safe to assume that once all of the server code is open then at first you'd need to roll your own apps pointing to the right server.
Since a lot of the interest for self hosting comes from companies then a nicely packaged version is somewhere in the future.
For an independent and serious comparison check https://www.securemessagingapps.com that focuses mostly on the security/privacy aspects of various messengers.
It's open source indeed [1] and was recently audited for the quality of the crypto protocol implementation [2]. Not sure what questions you specifically have but I'm happy to answer as I'm part of the team.
Open sourcing the client was always the plan after moving everything to E2EE. Just took a number of months to get the code into a shape that we wanted.
Monetization - we've invested 0 dev time into this so far. So a bit early to claim "they also failed".