I will try to answer these questions based on the fact that I buy most of my PC components online and I have built and shipped computers for remote clients in the past.
Question 1: I have shipped fully-assembled PCs in the original case package before, because this is what it fit in. Whenever I did this, however, I was careful to insure the shipment for close to or more than what it was worth, because I think in general this is a "good enough" solution and not anything close to a guarantee, especially the further you have to ship the product. In my case, I shipped products across one or two states this way for the most part without incident, using various shippers, but mostly USPS.
As far as buying stuff online goes - I have faith in this. I have purchased hundreds of parts online from Newegg, Amazon, E-Bay, etc. without more than one or two incidents of damage which were cosmetic in nature. Probably the most risky thing to buy online is a computer case - especially flimsier, cheap ones. These tend to respond poorly to harsh handling. You would think Hard Drives would also suffer due to their intolerance for jarring motion, but usually they are shipped with lots of protective bubble-wrap and I have not had a single failure that way. I suspect Many people blame hardware failures on the shipping process when in fact it was user error that caused the failure/misuse.
Question 2: I would package the items in a more robust external cardboard with extra tape. On the inside, I would make sure everything was buffered from this external surface by at least 1 inch of foam or a foam substitute, such as air pillows tightly packed.
Question 3: I would say it is always safer to ship computers part-by-part instead of assembled, simply because the interior of a PC is not generally designed to sustain torque of any kind, and I would worry about that at locations like the CPU heatsink mounting bracket and/or PCI-E interfaces with hefty daughter cards installed.
In order of least risky to most risky:
CPU
RAM
SSD
Motherboard
GPU
PSU
Fans, liquid cooling pumps, etc.
HDD
Case
I still generally consider it safe to buy all of this stuff online.
People have been shipping computer hardware for over 30 years. This question shows a complete lack of research on your part. – Ramhound – 2016-06-28T23:10:57.250
I was about to edit my question to take out the "Is computer hardware safe to ship" part from the question, but I decided to keep it because this might come up in someone else's research. And this is why I think it does not deserve a negative vote. – SpeedBirdNine – 2016-06-29T01:25:35.027
1If I feel a question isn't on topic, I don't find it helpful, awful habit of mine – Ramhound – 2016-06-29T02:19:27.400