Managing Office documents (content & permission) inside SharePoint 2013 document libraries

I have a document library inside my SharePoint 2013 publishing site collection. And I have the following scenario:-

1. Someone add a word document to the document library,
2. I can click on the word document inside the document library, and it will open inside my word client application:-
3. Then I can click on “Edit document” button inside the word document.

1. Do some modifications.
2. Click on “Save” button.

1. And the changes will be saved to the sharepoint document library (which is great).

But I have the following questions:-

1. Who can basically edit the word document and publish the edit to the SharePoint again? is this restricted to only users who have “edit” permissions on the document library,?
2. Can I specify that only me or certain users can edit specific documents?
3. If I open the word document using Firefox or chrome , I will not be able to edit the document and publish it back to SharePoint? So is this feature within IE only ?

Can anyone advice please?

Thanks