Thank you. They are changing everything now. I had to DEACITVATE IMAP in Gmail. (interestingly IMAP probably never was activated in the settings of Gmail, I just activated it, when the problem occured). After the deactivating the mails “returned” to my phone but not to my laptop. Now I found a “private” solution: I selected all mails in a category (label, folder), marked all of them as “important”, and I used the rule on the “important” folder in eM Client. It is a long procedere but it seems to work.
I just disabled the email scan in Kaspersky and it fixed the probem. Thanks for the idea.