Yes, you can sell and ship your items on Facebook Market Place. I have been a buyer and have had items shipped to me from different parts of the country. Facebook even reached out to me to see if I actually made the purchase with the buyer.
Because you have the option to accept PayPal I only used sellers who had that option available because of the buyer protection. However, I think it is best that you, as the seller, go directly to the horses mouth and read what Facebook guidelines are for selling on Market Place.
Of course all the communication was done on the Facebook Market Place messenger which I also receive a notification on my Facebook regular messenger. This all works quite well.
So click this link for a comprehensive discussion on the Facebook Guidelines for shipping items on Facebook Market Place where, I feel, most of your questions will be answered.