How do I sync Mail with Outlook and Gmail in browser?

I really like the Mail app in OS X, but I've noticed that it's not synced with the browser version of my email accounts, so anything I do in Mail (e.g. read an email, delete/trash it, etc) is only reflected within the app and not in my browser.

Just in case I didn't explain that properly: For example, if I read an email in Mail, and then switch to my Outlook account in Safari to check if that email has been marked as read, it would still appear as unread. If I delete and trash and email in Mac, that same email appears exactly where it used to be in Outlook, completely untouched.

This bothers me for two reasons. One, even if I were to stick with Mail and not touch my Outlook/Gmail accounts in Safari again, I still wouldn't like the idea of my inbox being all messy if I were ever to log back in. Two, Mail doesn't seem to have any options to create folders and subfolders as far as I'm aware, so my emails which are very neatly organized in Outlook are all lumped together in Mail's inbox. So even though Mail is a neat little app, I still have Outlook and Gmail bookmarked.

I'd like to know if there's a way to sync Mail with my email accounts in-browser, so that any changes I make in the app also show up when I log into my actual accounts? Maybe I'm just missing something or if this is not possible at all, please let me know. Thank you!