The http 404 error can be solved by changing "Options -> Select language..." from "Use System Defaults" to "English". Options menu can be accessed by left clicking your email address in right upper corner of the Launcher.^it seems this is Windows 7/10 issue for me, both Horizons accounts on Win 7 (Steam and that LEP-Frontier Store account) wont work at all, even in a base game mode.
But at Win 10 computer that other LEP Steam-account works despite that 404-page covering most of the launcher but "launch"-button.
With Win 10 it seems also that base game works without problems (bought from Frontier Store without extras).
E: also through Win 10 running just Frontier store base game with out Horizons/LEP, launcher also had 404 error but instead like with Horizons launcher also had 3 options to choose from below, not just "play/launch".
For the Steam related problems maybe the tricks posted earlier in this thread may help.