There is a lot of wrong information in this thread.
Gmail APP, doesn't let you pick how to connect it handles it all. If you have sync setup then it should delete it from all clients when they next connect/sync. This requires no setup what so ever.
I use Gmail app with two accounts, on A mobile...Tablet...PC...all of which sync. Including my offline gmail app on my desktop when connected to the internet. I get all my bookmarks, history and open tabs synced as well.
I also use exchange with another email account, which is a whole other selection of settings.
You also have the option to connect via POP/IMAP/Exchange but when using theses options there are client settings, and settings on the gmail website that affect what happens.