Rick Steves' Stainless Steel Water Bottle

Do you mean security? If without liquids, no problem.

But, since X-ray cannot tell if it is full or empty, you will probably have to take it ut of your hand luggage and demonstrate it is empty. Better to put in in your checked luggage.
Customs is nothing, just walk through the green channel. Unless you look suspicious, you will not be checked.

No. Just make sure it is totally empty and dry.

To avoid possible issues with an empty water bottle in the X-Ray machine, I take the cap off before security and put bottle and cap in my personal item. That way there's no question the bottle is empty.

I just put my empty water bottle in my carryon. I don't take it out, remove caps etc. I can't recall any inquiries about it.