I'm a foreign national, and resident in a different foreign country. I have no ties to USA, and no intention to hire any employees there.
Can I still register a company (e.g. C Corp)?
If it is possible and I do this via an agent in USA, how much will it likely cost?
[EDIT - I know there is likely a ton of other issues, like figuring out how to apportion income, dealing with tax, etc. but I'll deal with that bit later]
The total costs you are looking at are around $500 per year plus any mail forwarding costs and whatever a book-keeper will charge you. Since you said you are in Hong Kong, check out East West Bank. They are a Chinese bank with a physical presence in the USA. They might be able to open an account for your without you having to go through the trouble of nominating a CFO. http://www.eastwestbank.com/English/index.asp
Incorporation rules are different based on location (state). Some states are more strict than others and a simple Google search on this will point you in the right direction. That said, as a general rule, you will need a registered agent to act as the official business address in the state where you wish to incorporate. There are many services that offer to act as registered agents for people from other states and countries wishing to incorporate in a different state from where they reside.
Incorporation can range from a couple hundred dollars to several thousand. Registered agent services can be found for as little as a $150 a year and go up from there.