<p>In Sharepoint, is it possible to deny regular users the possibility of removing web parts and/or web part zones? If yes; how?</p>

The options to deny users from closing/moving a web part are in the 'Advanced' section of the web part's tool pane in sharepoint (Accessed via the 'Modify Shared Web Part' option in the web part's menu).
Alternatively you can modify the permission levels for the site (Site settings > Permissions > Settings > Permission Levels) and remove the 'Add/Remove personal web parts' and 'Update personal web parts' permissions from the levels your users run as.
Hope this helps.