How do you play a region-locked xbox 360 game?
You can play a region-locked xbox 360 game on a console for that region as you have already figured out. The three regions are generally identified by the different output standards that they support in those regions of NTSC, NTSC-J, and PAL.
Can an xbox account in the US region exist on a foreign console?
This seems to be your real question. You should consider changing the title. The short answer is yes. Also to address what seems to be your primary concern, achievements that you earn on your foreign console can be added to your gamertag, regardless of what region it is from.
See this thread about stacking achievements. The relevant post is as follows:
... So if I buy a PAL360(which is Europe)and get a copy of German Bioshock. All I need to do is attach my harddrive to the PAL360 and get more achievements for playing the same game, hence stackable. Is this correct?
Yes, but for the convenience of use you can simple store your profile in a Memory Unit and use it between 2 consoles, each one with its HDD.
You have to log with you normal account every time you play the games in different consoles, to earn the achievements.
So, simply put, you just copy your profile over to the other console either using an HDD or memory stick and sign in as normal. With the advent of the ability to have a cloud save of your profile, it may be possible to copy your profile over by simply using the internet, but I haven't read anything to confirm this yet.