Can I put modern LED bulbs in them?

Yes, modern LED bulbs are better for antique light shades than old incandescent ones. They run at a much lower temperature. Old tungsten bulbs got hot enough to crack delicate cased glass or scorch silk. An LED solves the heat problem and protects the fragile fittings.

Do old glass shades need special cleaning?

Keep it simple. Wash them in a sink of warm water with a mild dish soap. Never put them in a dishwasher. The heat cycles and harsh detergents will strip the surface finish and ruin any painted details. Dry them with a soft cotton cloth.

I have a brass lamp with a very wide fitting at the top, and I am wondering how to measure it so I buy the right glass shade to fit inside the screws?

You need to measure the internal diameter of the metal gallery. That tells you the fitter size required for the shade. Loosen the three screws on your fitting and measure straight across the opening. Standard sizes are often two and a quarter, three, or four inches across. The neck of the glass shade must be slightly smaller than this measurement to slide in easily.

My grandmother left me a silk fringed shade that has got very dusty over the years, is there a safe way to remove the dust without ruining the fabric or the trim?

A soft brush is the safest tool for old silk. Work gently downwards from the top ring towards the fringe. Vacuum cleaners have too much suction for fragile antique textiles and will pull the weave apart. If the dust is stubborn, a hairdryer set to a cool low speed can blow the loose particles out of the fabric.