yes, you can mail cuttings. i've done quite a few trades with cuttings as well as seeds - both coming to me and me sending out.

with cuttings (or stuff that's already got some roots going) you should wrap the root area in moistened newspaper or paper towel and then cover that with plastic wrap, tinfoil or a plastic bag that's secured with a twist-tie or rubber band. you'll want the item to arrive in good shape so you need to keep the moisture there.

with cuttings, i've used bubble wrap envelopes or boxes. i've also, in a pinch, used a regular envelope and cut pieces of heavy cardboard to fit in the envelope to protect the plant from being crushed.

it's good to mark these types of packages as 'live plants' and send them the quickest way possible...anything that is going out of your state should go by priority mail anything in the same state can go first class...it's close enough that they'll get it within 2 days (why pay extra fees for a service if it's not needed?).

there are restrictions when sending live plants to other countries...some are acceptable, others aren't. some countries don't allow even seeds to be mailed in. you can get that info off of the post office site usps.com