"So if I do not have an Internet connection the messages on an IMAP server are not shown. An extremely big drawback, I would think. How can I get those messages to the drive / download them to the drive like it is done with POP e-mails? So that I can open them without Internet connection?"
Offline folders are enabled by default. If you are working online it shows the contents of the remote folder. If you are working offline (need to select file -> offline -> work offline, not just have no Internet connection) it shows the contents of the offline folder if it exists.
Personally I've never felt the need to read my mail using Thunderbird when there is no Internet connection since that mainly occurs when there is a power failure and I can always fallback to webmail on my smartphone for my main account.
"So I would have to find and just directly copy that mbox file only while I am online to back it up? There is no such option in Thunderbird?"
You can export folders as mbox files or .eml files using the ImportExportTools NG add-on. The add-on also has support for backing up the profile. Long term they plan on migrating that add-ons functionality into Thunderbird, That add-on has other export options, but it only knows how to import mbox files and .eml files.
One downside is sometimes when there is a new major version of Thunderbird it may take a few months until the add-on has been updated to support the new version. That's one of the reasons why they want to merge it into Thunderbird. That problem is not specific to that add-on.