On my desktop application I am able to access a drop down of each parent item. However, on the online version, it is not giving me that option. Please advise.
Web applications don't have context menus (generally) — the browser just shows its own context menu. Functionality that exists in the web library is available in the toolbar after you select items. But many features are only available in the desktop app:
Thank you! I am also curious if there is an option to have a PDF open and take notes in the note feature on the online version. Or do these two things have to open in separate tabs?
Correct, the reader's right-hand pane — which you can open to a Notes tab to take notes alongside a PDF — isn't currently available in the online version.
You can create PDF annotations (including sticky-note annotations) in the online reader, though.
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https://www.zotero.org/support/kb/web_vs_desktop
You can create PDF annotations (including sticky-note annotations) in the online reader, though.