In general, you have to use the desktop environment to start the window manager, not the other way around; the window manager must be controlled by the desktop manager's session manager. The xmonad configuration you were given just minimally configures monad to play well with most desktop managers, but without any integration beyond using the LXDE terminal; in particular, it won't configure mod-shift-q to shut down the LXDE session, and the session manager will usually just restart the window manager if it exits so mod-shift-q will end up being the same as mod-q.

Your xmonad.hs is the one from section 3 of https://www.haskell.org/haskellwiki/Xmonad/Using_xmonad_in_LXDE but you also need the file from section 2 to configure LXDE to start xmonad; then you use "startlxde" as normal to start an LXDE session with xmonad as the window manager. I would use a slightly more detailed xmonad.hs to fix the mod-shift-q issue, though:

    import XMonad
    import XMonad.Config.Desktop
    import XMonad.Util.EZConfig
    
    main = xmonad $ desktopConfig { terminal = "lxterminal" }
                    `additionalKeysP`
                    [("M-S-q", spawn "lxsession-logout")
            

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