Can't save files in some applications
Since you haven't used ls -le as recommended, we don't know what the access control list looks like, but there might be something going on there that is preventing you from being able to write the files. The fact that a newly created user CAN edit the files makes me even more suspicious.

This command will remove ALL access control list entries from the folder and all its subfolders and files:

sudo chmod -RN path/to/folder/containing/files


Once you've cleared the access control list, try resetting the ownership like so:

sudo chown -R ulrikdamn:staff path/to/folder/containing/files


Now grant permissions like so:

sudo chmod -R u+rwX,g+rX,o+rX path/to/folder/containing/files


This gives you full read/write permissions. The "staff" group and all other users get full read permissions. Using an uppercase X, sets the execute bit for all directories within the tree, but leaves the execute bit for regular files as is. 

Lowercase x would set the execute bit for all folders AND files. The execute bit must be set for a directory in order to read its contents. If you'd like to give write permissions to the group or others, switch out for g+rwX or o+rwX as necessary in the last command.

Here I'm passing in a folder path. If you want to pass in a single file, just remove the -R from each command. The R makes the command recursive, applying it to an entire directory tree. Without it, the command will modify the permissions for a single file or folder passed in.