Nut size (A.K.A. diameter) is the size bolt that the nut is used with. The same is true of the thread count. Thus a 1/4"-20 nut fits a 1/4"-20 bolt.
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10-32 is a fine thread, but it is listed under coarse thread because it is very common.
NOTE: While 12tpi is the fine thread (UNF) standard, 14tpi is far more common in actual use.
Thread locker prevents fasteners from loosening after installation. Anti-seize helps prevent seizing (galling) during installation.