Organs are typically left in place, unless the person is an organ donor. If an organ donor, machines will keep circulating blood through the body until the body is taken to a surgical suite where surgeons remove only the organs or skin that is usable, including corneas (outer film over eyeball).
Organs receive embalming fluids as it courses through the body. Therefore, organs do not decay or decompose like they would in a body left without embalming fluids.
Only if you are going to be embalmed.
Afterwards his vital organs were removed during an autopsy
Neither. Both the heart and brain are vital organs, and both are neccessary to keep your body functioning. If either of them were removed in any way, and not instantly replaced back in, you would die within seconds.
because the brain was a organ that needed to be removed and put in a canopic jar which is where they put all the organs
Surgically.
lungs,
After the internal organs have been removed the body decays
All major ones are removed.
Presuming you are talking about the modern world, no. In fact, as a rule no organs are removed from the body, unless you consider the blood an organ. All or most of the blood is removed. If there is an autopsy before embalming, then any number of organs might be removed depending on the pathologist's needs.
large and small intestines
If the stones are blocking important and vital organs then yes they will need to be removed. If not its still a good idea to get them removed because you never know when they will start blocking vital organs.
Of course they can, if your organs froze you would die!