Hi Christine…and Jigs. I finally took a look at Project Plan 365 and it seems to be a nice, simple tool. It’s an alternative but doesn’t answer your question with Project Online.
To answer your task list question, you should be able to just add the Task app to the home page (assuming that’s what you mean by the entry screen). Just go to the Home page, select Edit Page, add Web Part, and select the Task app. The home page can be configured to display any of the site content.
Your question on using Team sites is more difficult. I’m loosely familiar with team sites so my answer here may need additional verification. Team sites by default are setup as site collections and doesn’t work with Project Online, at this time anyway. Teams is Microsoft’s future collaboration solution, and will have to work with Project Online eventually (my opinion only). If I’m correct and seeing Microsoft’s progression with Teams I would assume sooner than later. Anyway, future direction at best. In the meantime if you are ready for teams, you might want to try a team site per project and configure quick launch links to go seamlessly back and forth for the project schedule. I haven’t done this but could work if that’s what you need.