Strange sync with account

Having account, and the IMAP in eMclient (version 7.1, Win10) has 2 problems:

  1. when I open eMclient and I start reading e-mails and move them to corresponding folders, some e-mails disappear from inbox, there are on the web - the syns does not work. The only solution is restart eMclient app. I  have not found any “rule” which e-mails are hidden and why

  2. if the eMclient is open, and eg. in Android I read the e-mail, the e-mail in eMclient is still unread (on web is seen as read), does not matter if eMclient is restarted or not. Sync does not work in this case either.

PS. in MS Outlook and Android Gmail clients everything works OK, so the problem is on eMclient side.

PS2. Thinking about to buy full license but above mentioned problems are quite crucial for me.

Sounds like you are using POP3 and not IMAP.

Both account are set to, port 993

eM Client uses IDLE IMAP (PUSH IMAP), so you need to check with if their IMAP server supports it.

OK, does not support PUSH IMAP.
But IMHO it does not explain why some e-mails disappear from inbox when they another e-mail is moved to local folder.
And if I manually asks to refresh inbox, it should at least in this way. And it does not.

Refresh will not sync IMAP. You will need to close and restart eM Client for it to sync with non-PUSH IMAP servers.
You have only two options: use POP3 or another email client.