Open 0xr.space once with a connection. The site keeps its main files, and the parts of it you have used, on the device, so later, without a connection, the page can still load and the models already stored there open from their cards. For an icon, use the browser's own Install or Add to Home screen option, and Model Explorer opens in its own window without the browser bar.
Offline is a fallback, not a mode. The page always tries the network first, and after the site updates, only the parts loaded since then are kept, so a visit online after an update is what makes the next offline one work. The Gallery, its search, Nearby, share links and Photo to 3D need a connection. There is no install button in the app itself; on Meta Quest the free Model Explorer app on the Meta Horizon Store is the installed option.
No. It runs in the browser. Installing only gives it an icon and a window of its own.
The Gallery and its search, Nearby, making a share link, opening someone else's link, and Photo to 3D. Viewing a model already on the device does not.
Only if the reader for that format was loaded since the site last updated. Otherwise it needs a connection to fetch it first.
Yes. Model Explorer is free on the Meta Horizon Store, and the Quest browser opens 0xr.space as well.