Yes you can get a stomach transplant although they are uncommon and usually done along with other organ transplants such as the pancreas or intestines.
There is no evidence supporting a procedure where a pig's stomach was actually transplanted into a human. However, there is research going on involving a procedure like this.
Yes they can be transplanted.
Yes, livers can be transplanted.
Routinely transplanted organs are: heart, kidneys, liver, lungs, pancreas, small intestine and cornea. Other transplants are available, but are not routine. Some organs are just not routinely transplanted because you can live without them, e.g stomach, bladder, spleen, colon etc... (Yes, you can live without your stomach, it's only the small intestine which is necessary for food absorption.)
Its like any other transplant you hear of, except with a stomach. Doctors surgically remove your stomach and replace with another healthier one.
Qui transtulit sustinet (Latin "He who transplanted sustains", also "He Who Transplanted Still Sustains" or "He Who Transplanted Continues to Sustain")
Just to clarify your question, if the organ cannot be transplanted there is no "recipient", since nobody receives the organ. The brain is not transplantable in humans. It has been performed in labs on animals, with mixed outcomes. However in humans it is unclear whether there are any medical conditions which would actually benefit from a 'brain transplant', let alone other ethical questions of performing such a procedure. The large intestine is not commonly transplanted because it is not necessary to sustain life (however it is medically possible). Likewise, the stomach, gall-bladder and appendix are not necessary for survival.
The cornea is the normally transplanted part of the eye.
He who transplanted still sustains
The first successful stomach transplant was performed in 1959 by a surgical team led by Dr. Ronald Lee at the University of Minnesota. The recipient was a 3-year-old girl who had her stomach removed due to a tumor. The transplant was a groundbreaking medical achievement at the time, paving the way for further advancements in organ transplantation.
no
Hearts, liver, kidneys, lungs, small intestine, pancreas, corneas, heart valves. Some other things have been transplanted (both successfully and unsuccessfully), but typically only "life saving" organs are routinely transplanted.