Can I put these in a modern dishwasher?

Do not put antique plates in a dishwasher. The harsh detergents degrade the glaze and strip away overglaze enamels. High heat causes sudden expansion. This stress cracks an older clay body. Hand wash them in warm water instead.

I have a stack of old dinner plates with lots of brown staining under the surface and I wonder if there is any safe way to clean them?

That brown staining is caused by moisture and oils seeping into the porous clay body. It happens when the protective glaze develops fine cracks over time. Soaking them in a weak solution of hydrogen peroxide sometimes lifts the stains. It takes time and patience. Make sure you rinse the pieces thoroughly in clean water afterwards to stop the chemical action.

What does the printed diamond mark mean?

That is a British registry mark. The Patent Office issued these to show a maker had registered a specific design. The letters indicate the exact date of registration.

Sometimes I see metal staples holding a broken plate together and I want to know why someone would bother fixing it like that instead of throwing it away?

Ceramics were expensive commodities before the industrial revolution. Replacing a single broken piece from a large matching dinner service was often impossible or costly. Travelling riveters repaired these pieces using a hand drill and brass wire. They drilled partway into the ceramic body to insert the metal staples. The tension held the shards together. These historic repairs are now part of the object's story. Many collectors prefer to leave them exactly as they are.