The Access/Sharepoint option won't help you at this point. It would require a complete rewrite of the application to use Access web forms. Web forms are severely restricted and don't support all the events that standard forms support and they don't support VBA at all. You would need to rewrite all your code as macros and macros may not have the functionality you need. For example, you need a real programming language to automate Word and Excel. Macros can't do it.
Why do you need to share over the internet? Do you have remote users in other locations? Using Remote Desktop doesn't require that every user connect that way. The local users can run the FE from their own computers and link to a BE on the server and the remte users can use Remote Desktop so you may not need as many connections as you thought. Remote Desktop/Citrix is the best solution I have found for sharing Access applications. You could look for an ISP that provides Citrix support and let them run the Citrix server for you. It probably won't be cheaper than doing it yourself though.
1. I think Office365 is Word/Excel/Outlook running in a web browser rather than your local hard drive.
2. If your data is stored in Sharepoint tables (I don't recommend that solution but it can work for a simple app), you can have the Access FE loaded on each computer link to sharepoint tables. If users don't have Office Pro, you can download and install the Access 2010 runtime which is free. The A2010 runtime will run earlier Access version apps but when using the runime, my recommendation is to develop in the newest version you can. You will only need a couple of full Access installs but all the users can work with the free runtime.
3. I don't understand the question. Access isn't running in the browser unless you are using webforms.
4. Again, I don't understand the question. Sharepoint has security that you can use if you go with Sharepoint lists.
5. No.