Yes, SharePoint 2013 has Visio Services which allow SharePoint to display visio diagrams effortlessly (on desktop AND mobile) - SharePoint 2010 not so much without heavy custom coding
Not at all. A visio diagram can be saved normally, just the save location needs to change (to a library in SharePoint)
Definitely. Since visios can appear directly in the browser, you can display them however you like.
Visio documents are still documents - search will display them without issue. You can enhance the end result display as well.
Using Visio Services, yes.
Yes it is best, but not necessary. Publishing directly to sharepoint requires 2013 I believe (could be wrong here)
It will be. You will need to customize extensively to do what you want without end user adoption issues.
I have no experience with AWS sorry. I would think you would go with Office 365 or something though, if that is an option.
Not many if you have SP2013 with Visio Services enabled and are working with Visio 2013 (professional or premium edition). If you have this, then the end user 'front-end' is all you will need to really worry about. How it is displayed and interacted with by the end user will all need to be determined.