When is the correct time to place the base board, before or after placing split brick?
Also, is quarter round normally placed when the floor is tiled
A 7/16 x 3/4 +/- "shoe" molding (less often a quarter round) is often used when the baseboard has already been installed or if the detail is consistent with the house design and historic period.
Is that a 1"+ thick kind of paver/tile?
If so, there's no perfect answer; the tiles vary, and are never set to a perfect line at the perimeter, so it's always a low-down dirty job to scribe the woodwork to it, but it's IMO uglier to grout up to already-placed baseboard.