I found the Thunderbird remindIt plugin.
From the send button you can open the dropdown and puss "send and remind". It will automatically tag the mail in the sent folder with a
RemindIt.Pending tag. When somebody answers the mail, the tag is automatically removed.
So far that is absolutely what I need.
There is a reminder text to be configured, so it looks like it will re-send the mail with a reminder text after an also configurable reminder timespan.
In my tests I could not see this function.
But what happens is, that the tag of messages that were not answered changes to
RemindIt.Expired so you can make a search folder based on this tag to see which mails need answer.