There is an action (see FAQs on how to create an action) to import cover files. By default this imports a matching file as "front cover". It is not possible to set the cover type during/for import.
There is also an action for exporting covers.
So now, for an automated approach:
Export with an action the covers from all mp3 files, may they contain "front cover" or "other". The first cover is exported by default to a file called folder.jpg.
Then import with an action all covers again. they will be set to "front cover".
As various programs treat the tag data in different ways there can be no general advice on how to get updates across to these programs. As far as I know do iTunes, VLC and WMP maintain a separate database which has to be updated. Esp. iTunes becomes more and more reluctant to recognize changes in the files in the younger versions.
The only thing I can say is that files that have been tagged by mp3tag are tagged according to standard and the covers should be visible in any program that follows the mp3 standard.