Having just completed my first project using Google App Engine, I can say that I ran into alot of the questions that you have. I'll try to explain my approach to each point and also approach it from your perspective as well.
Authentication - Generally using Google's auth would be the easiest route, but you would still have to implement a custom adaptation in order to work with the "company"/"group" concept. Implement in the datastore/whatever database you prefer to use an entity called "Groups" which have a list of google users... this way users can belong to many groups.. then you just search by property (user) to get all groups they belong to. I implemented my own authentication system for unrelated reasons.
Google App Engine comes with SSL/HTTPS support for its own domains. You can add in your own custom domain with SSL support as well. You can use SSL through native apps or mobile web apps additionally. I simply used the native support that came with it.
Yes and no. You will always have to store the credentials somewhere. Maybe it wont be in your apps code/directly connected to your app (Google auth would be an example). But somewhere, on your phone, the credentials WILL reside. They may be encrypted/obfuscated, but they will be there. So either have your user enter them in everytime, or save them/use the ones provided by the phone. For myself, .NET provided a nice way of storing credentials in a secure fashion (non-plain-text) for each user's machine.