which countries sing happy birthday to the same tune as in english (or
alternatively, if the list is shorter, which countries do NOT)?

There seems to be several different songs with composers as varied as
Alfred Schnittke and Pyotr Il'yich Tchaikovsky, to name but two.

Arguably, the most popular version is by Mildred J. Hill and her
sister Patty published in 1893 in their book Song Stories for the
Kindergarten, calling it ?Good Morning to All.? However, the Hill
sisters didn?t compose the melody all on their own. There were
numerous popular nineteenth-century songs that were substantially
similar.

You may find this article of interest:

http://www.boycott-riaa.com/article/15999

So, it doesn't seem to be easy to answer.

It is pretty common in Germany, unfortunately, I feel, since there are
German songs that can be sung as canons that congratulate with more
thoughtful content.

More than ten years ago, I heard a Chinese group in a hotel restaurant
in Irumchi (NW China) sing it in Chinese.

I've personally heard it sung in German, French, Spanish, English and Esperanto.

My family always sings the first verse of 'Happy Birthday'. It's a
'real' first verse, yet I have never been able to find any reference
to it -- but we don't sing Happy Birthday without it!

We sing the Happy Birthday song in Turkish as well.