Remote web application accessing SharePoint with current user's credentials

Question

Can we access SharePoint from a remote application with current user's credentials when

1. user opens a remote page which has server-side CSOM
2. user opens a remote page which has a server-side WCF client for SharePoint
3. user opens a remote page which has a server-side Web Reference for SharePoint asmx services
4. user calls a WebAPI REST Service which connects to SharePoint (with 1, 2 or 3)
5. user opens a remote page which uses JSOM
6. user opens a remote page which connects to SP REST services via JavaScript

Should these easily work in on-premises Windows Authentication, Kerberos and O365, and what we need to take into account?

Answer 1

Pretty sure the answer is Yes to all of the above. 4,5 and 6 best suited to O365 and the rest on premise.