I have figured out how to copy and paste whole cells, but not any kind of text (from clipboard). When I right-click in a cell (in the web browser), the 'Paste' option is shown, however nothing happens!
I found no problem with Copy|Paste in Mathematica Online (I'm on OS X). It worked allright selecting a single word from a text cell and using all possibilities: right-click|Copy, Cmd-C, and Edit|Copy, to Paste it into the same cell or in some other cell. It works fine. Same with pieces of code from Input cells.
I have found that this doesn't work in Safari (as of 12.0.2), but it does work in Chrome.