Is there a way to insert some data into a range; shifting the range down after inserting?
You need to do the shift first and to paste/enter/whatever your data/formulae/whatever subsequently.
You didn’t name your LibO version (Please always do!), and the UI may have changed insofar within the last year.
With recent versions select the range you want to shift and go > Sheet > Insert Cells and select the mode.
LibO will refuse to shift non-empty cells off the sheet.
Using Paste Special… (Ctrl+Shift+V) you also get offered the option to perform a preparing shift operation in advance by the same command. Inpermissible shifts will then not be offered for selection at all.