Is it possible to set List-permissions in a way that just single Persons can edit columns from ListItems?
To be more specific i want to approve (setting a checkbox "approved") Listitems after checking them and this check hands those Items over to another Listview (Approved Items)
The requirements you presented are very well covered by the Out Of The Box Sharepoint Content Approval mechanism. It provides you the possibility to Approve/Reject items and gives you a view of approved/pending/rejected items, grouped by their status. If that view is not what you are looking for, you can always create your own one, just filter on the "Approval Status" column.
So, to enable this feature of Sharepoint, go to List Settings and in the General Settings, in the first column you will find Versioning Settings. Open it, and choose Yes for "Require content approval for submitted items?" For a simple scenario keep No for "Create a version each time you edit an item in this list?". But of course you can enable this too, if you want versioning. Choose "Only users who can approve items (and the author of the item)" from "Who should see draft items in this list?", because I think this is what you asked for in the requirements.
After you click OK, a new column appears, Approval Status, and two new views: Approve/reject Items and My submissions.
In order to restrict who can approve items, put those people in the Approvers Sharepoint security group. To access it go to List Permissions. If your list inherits security settings from the site, then you will add these people to the site's Approvers group. If you have broken inheritance for the list, then whoever you put in the Approvers group will be approver just for this list.
I hope this was what you were looking for. :)
The smallest unit for which permissions can be set is SPListItem. If you need to set permissions to field you can try to use one of the existing open source solutions. For example http://spcolumnpermission.codeplex.com/.
