1) They will remove their shoes and PAD AROUND in their socks
Better:
They will remove their shoes and WALK AROUND in their socks
or
They will remove their shoes and MOVE AROUND in their socks
2) In a hot, humid environment like Japan’s, food GOES OFF quickly
Better:
In a hot, humid environment like Japan’s, food GOES BAD quickly
or
In a hot, humid environment like Japan’s, food TURNS BAD quickly
or
In a hot, humid environment like Japan’s, food SPOILS quickly
3) In the room where the tea ceremony is HELD everything is absolutely clean
This is correct, and it is the best way to phrase this sentence.
4) So why aren’t all Buddhist nations as ZEALOUSLY clean as Japan?
That's pretty good, but if you want a simpler word:
So why aren’t all Buddhist nations as OBSESSIVELY clean as Japan?
5) Said to ENSHRINE the very soul of the Japanese identity
This is good too, but if you want simpler:
Said to CONTAIN the very soul of the Japanese identity
...Hope this helps. Good luck!