Are other Libre applications running in the background when opening Writer?
No - unless you have opened them.
Well, there is actually one option to kinda have LibreOffice in background: use an old version with the quickstarter.
Yet this doesn’t make “other applications” run when opening Writer: in that moment, it’s only a single LibreOffice process in memory, with Writer DLLs loaded, and no other modules active (only a single icon in taskbar compared to operation without quickstarter) 
Thanks. Since I’m only using Writer, I don’t want to bloat RAM with extraneous application data.