What's the best thing to use to fill some gashes in an older wood floor? Wood putty and then stain?

what i would do is find a piece of scrap wood of the shade you need and try to cut or splinter off a segment that looks like it might fit the "gash". use some carpenters glue and set it in place to let dry. after it's set some light sanding will bring it to grade. then if it needs it you can apply some polyurethane.

A rolling chair doesn't usually leave a gash, it's more of a dent or a trough; and they're usually too broad and shallow to fill in. If it's near an edge you could replace a few of the boards and stain/finish them to match, but chair marks aren't usually near an edge which makes replacing boards difficult. Another option would be to have the floors sanded and refinished.

Crushed wood fibers can sometimes be "re-inflated" with a steam iron, provided there's no varnish or other finish that impedes the flow of steam into the wood.

Set iron to 'Steam'> rub back and forth over the groove. It takes a while, but does work; plus you won't have bonding and/or staining issues if it works.

After the fibers have swollen, you might want to coat with the finish that closely matches the balance of the area.

A good quality acrylic vinyl tube caulk for any joints and painter's putty ( for painted trim nail holes) wood putty of appropriate color for stained trim. Available at paint departments everywhere.

Try rubbing a crayon over the superficial scratches. By melting the wax, it will fill in the gashes. It will stick better than putty and it wont crack.