I've found tiny brass screws at Mcfadden-Dale Industrial Fasteners in Santa Ana, CA for about a nickel each.
Most industrial fastener places should have what you need. Google it and find one in your area. A good shop will have every type of nut and bolt you could ever need, for pennies.
To get old screws out, I make sure the slot is clean of paint and rust, usually just by running a knife blade thru it, then I use a small snap-on pocket screwdriver I've owned since working at a gas station 25 years ago.