How can I hammer finishing nails into molding without hitting/chipping

Use a nail setting tool

Is it possible for you to borrow or rent a finish nailer ?

Get a "nail punch" at the hardware store. Put it over the head of the nail when you're almost in. It pokes the head down into the wood while protecting the molding.

Nail in to about 1/8" to 1/4" above the surface. Use another finish nail on head of the nailed in nail and hammer to countersink it. Finish nails have a divot in the head for a nail set but this will work if you don't have a nail set.

you could also get a bit for your drill to countersink a trim screw

You could invest in a pneumatic nailer. If you have much to do and plan future projects it makes all the difference. Big box hardware stores are offering a compressor with stapler, finish nailer and construction nailer for $200. The quality of your projects will improve dramatically. Good luck