If there's probable cause and a warrant, sure--just like there should be anywhere. But no, warrantless mass surveillance of mosques just because they're mosques would be a significantly unconstitutional overreach.
Seems like a reasonable statement.
It is, with the one caveat that sending in informants doesn't require a warrant generally. But the idea of widespread monitoring of mosques is a bad idea.
Oh, of course. Obviously anything a mosque puts out to the general public wouldn't require a warrant.