How to display files in a web browsers?
I'm not aware of any web based file browser that "opens files in the client site applications" like Project, Word, etc. and saves them.

Some ideas:

However, this could be possible if you open your files with a FTP manager, as they will be opening from your client's computer, therefore using its applications. 

Another option might be the download / re-upload approach, for what you could use a module like elfinder, which does allow you to edit text based files (note that we are not talking about propietary software like Microsoft Word).

Disclaimer: I'm not endorsing any of these methods, just offering alternative ideas.