how to clean brass drawer handles?

Take them off the chest first. If you polish them in place, the chemical cleaner will stain the surrounding mahogany. Wash the brass in warm soapy water to remove surface grime. Most dealers leave the darker tarnished spots alone, as bright shiny metal looks harsh against old wood.

The drawers on mine are stiff and hard to pull out, especially in winter, so what can I do to fix that?

Wood expands when the air gets damp. Central heating dries rooms out in winter, but if your chest sits in an unheated room, the deal carcase will swell and grip the drawers. Rub a plain wax candle along the bottom edges of the drawer sides. The wax acts as a dry lubricant and helps the wood glide over the runners. Avoid using liquid oils.

missing keys for old locks?

A traditional locksmith can cut a replacement. They take an impression of the internal levers to match the original shape. You can also buy batches of old keys online. Often one out of a dozen will happen to turn the lock.

My grandmother left me a chest and the top has these dark circular marks where somebody must have left plant pots sitting there for years, can they be removed safely?

Those rings are water stains that have soaked through the polish and entered the timber. Plant pots constantly weep small amounts of moisture. Surface wax remover takes off the top layer of dirt, but deep dark stains usually mean the wood fibres are dyed. A restorer would need to strip the top and bleach the marks out. Most people just leave them. They map the history of the piece.