Xbox 360 not logging in correctly

As far as I know, there's no direct way of getting this information. However, you can safely go ahead and make your purchase anyway. Since downloads are associated with your account, they'll appear in the Active Downloads section of the Marketplace pane on the Xbox button menu no matter which Xbox 360 you log in from.

If the current Xbox 360 is not considered your primary console, the item in the queue might indicate that it's "Assigned to other console." If this happens, selecting the item and pressing A should bring up an option to download all items. Choosing to do so should update your account to consider the current system as your primary Xbox 360, and downloads will happen automatically when you log in from that point forward.

Microsoft has a Knowledgebase Article on this topic that might also contain relevant information, and Major Nelson did a blog post when this first became a feature.

I'm not sure if it's necessary to do it manually if you go through the above step, but you may also need to transfer licenses for any content you had originally downloaded on your other console.

To perform a license transfer, on your console, go to the Settings pane. From there, click on Account, and select the License Transfer option. As you can only perform this action every four months, it'll ask you to confirm. Doing so should transfer all of your existing licenses over to your new Xbox 360.

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.