Will backing up using time-machine also backup all the users?

Time Machine backs up an entire drive/partition/volume unless you specifically remove files and folders under "Options". All User accounts are included unless you uncheck them in Options.

All of this info is available in your Finder Help menu.

In essense, it's not a true one to one backup, but it's like 99.9% there. It will backup all your files, users, user documents, settings, applications etc, but not quite every file. Pagefiles, certain cache files, logs areas etc can be ommitted as they are recreatable and as such don't need to be recoverable. I don't believe you need to be an admin user to do a backup, as the backup daemon is always running as a priveleged process anyway and you just pass it the message to go and do it's stuff, therefore you do not need access to all the files under your own accounts permissions.