I saw you post this on ask ubuntu too (https://askubuntu.com/questions/1195648/panel-doesnt-stay-on-top-in-lubuntu), but I have limited experience with appimages.
Appimages don’t use your own icons (have their own icon set they use packaged with in the appimage itself), and I suspect it’s similar with knowledge of the panel (they don’t know of, nor respect what you have selected). I noted on github there was no desire to change the icons issue, and whilst I couldn’t find anything on panels - I suspect it’s the same.
I would personally use the default deb packaged version (https://packages.ubuntu.com/eoan/libreoffice), or the snap version
guiverc@d960-ubu2:~$ snap search libreoffice
Name Version Publisher Notes Summary
libreoffice 22.214.171.124 canonical✓ - LibreOffice is a powerful office suite including word processing and creation of spreadsheets, slideshows and databases
projectlibre 1.9.1 jibel - Project Management software - alternative to Microsoft Project
Maybe another user with more experience (with AppImages) can provide the answers you seek, sorry I can only offer alternatives.