Is it possible to read mail without entering the mailbox. Just click on the account and the Inbox will open on the right, and on the left the account will be collapsed. As an example made in the screenshot.
No, you need to click on the folder in which the emails are located. The account name is not a folder and does not contain anything to read.
You can create a Smart Folder to display all messages in all folders for that account, if that helps.