Hello!

I believe that this happend with version 11.0.6 - when starting Conversation with someone this Conversation then appear in Messenger having a title of a few starting words from Conversation but the Chat window is empty. Is this normal, or?

Yes, now when you create new conversation or if talk doesn't contain at least one message there is will be Empty message.

I would also like to suggest a feature:
Possibility to save a chat history. If someone is chatting with someone they would both have possibility to save - download the chat history....

Good idea, I have added it to the list https://github.com/unaio/jot-client-una/issues/140