any stone safe cleaner will be fine. mortar is just a cement based mixture.

Have you applied a sealer to any of the bricks? maybe clean them, let it dry, and apply a penetrating sealer that won't change the colors?

this is safe for marble and any stone. but always spot test. some of the other masonry cleaners are acidic. but I don't think you need that.

Nothing will hurt the mortar a scrub brush and any household cleaner will be fine and I have no idea why sealing is nessessary. I would however change that country style gate.

because she says it's beginning to stain. the sealer makes the top mostly waterproof and keeps the brick from further staining (either from water deposits or whatever else).

You can try the following steps:

1. Mix a tablespoon of bleach with a gallon of warm water in a bucket.
2. Apply the bleach solution with a sponge.
3. Enlist a bristle brush to scrub from the top down.

Thank you