Yes. With the possible exception of "face transplant donors" nearly all organ donors can have an "open casket" funeral.
Open casket possible. Organ donation requires opening of the abdomen and chest through a surgical incision. The heart, lungs, liver, pancreas, intestines, and kidneys are all recovered through one incision. Donation of tissues like corneas, bones, skin, veins, tendons may require different incisions but are always accomplished with an open casket funeral as an objective, and prosthetics help preserve the appearance.
Yes. The body is surgically closed after solid organ donation.