Interesting problem.
Two solutions:
1 - use the autoinstaller to update. Be aware that this can cause some data loss in your case, if configuration is also somewhat messed up.
2 - Manually change ownership.
The nice thing about your problem is that only webmail and qmail are not owned properly.
Qmail can be easily resolved and set back, by a combined use of both solutions. Follow the steps:
1 - Open command line utility (SSH)
2 - run /usr/local/psa/admin/sbin/autoinstaller
3 - at first, only select Postfix at "Mail Servers" and install
4 - when the installer is finished, open the autoinstaller again (!)
5 - now select QMail Mailserver at "Mail Servers" and install
Problems with QMail should be resolved.
Note: if you have postfix as your mail server, then simply first select qmail and then reinstall postfix.
IMPORTANT: when using qmail as a default server, you should manually change the ownership of qmail-local. See:
http://forum.parallels.com/showthread.php?t=94593
http://forum.parallels.com/showthread.php?t=87343
The webmail issue is somewhat different since that can involve two webmail systems (horde and atmail) and i do not know what you use and what the problems are.
After all, as far as i know, horde is supposed to work under the root:root ownership structure. So, there is no mistake over there.
In short, first try to reinstall your mail server again.
NOTE: if the reinstallment does not resolve your problem the first time you run autoinstall, do it again (!). Due to a minor bug in the autoinstaller it can be the case that the change between postfix and qmail is not executed properly at first. Since files are left on the file system when changing, a newly executed change is very likely to install all missing files (the files that resulted in the change not be succesfully the first time).