Thanks Peter for sharing this link re the first point.
Re point 2: No, there are no <p> and </p>'s. I know that this will result in proper showing, but the users are 'simple minds' and I don't want to bother with adding this to their text. So they just enter the text in the (classic) Visual editor.
But now it shows as a concatenation of words, all on 1-2 lines.
For this case I'll fall back to the display-posts plugin that handles it out of the box and also respects <!--more-->.
Thanks for your help.