You can ship almost everything. It all depends on how much money you have to spend on it.
We've moved two and from the US twice, and here is our experience:
Any personal/household items - yes, the who lot. The only thing to leave out is booze if you don't want to pay duty (and more importantly a customs inspection of your load), and anything that's considered hazardous.
Bicycles - no probls
Cost is dependent ont he size of your container (or the amoutn of the container that you are sharing) for sea shipments, and the weight for air shipments,
Yes, door to door is normal. The folks who do the sea shipment will have agents who do the land parts. You shouldn't have to worry about this.
Timings: we found that translantic was one week from port to port
Missing: no, short of a major catastrophe. You wax-seal your container, and it's not opened again unless customs in your destination needs to open it.
There are specific standards for international shipping. Everything is individually wrapped.
Other things to consider is how will you live while your goods are in transit. It takes 3-4 days to pack up a house for international shipping, and you are pretty much out of the house on the first day. If you are going direct to your destination, you need to either stay in a hotel, or rent furniture and kitchen goods to tide you over for a few weeks. There are companies that specialise inthis kind of rental.