Why not show some repect to the old man and show is in the plural form as for Julia?
Definitely. To be polite, you should use 'बूढ़े हैं'.
(Also, the word बूढ़ा itself is not always very respectful, even though it is quite a common word. बुज़ुर्ग is a better alternative).
should you use ये instead of यह as well?
Yes, if you're using the respectful form with बूढ़े हैं.
I guess it's just written to say both are accepted.
"यह आदमी बूढ़े हैं" still not accepted.
The sentence using the respectful plural would be ये आदमी बूढ़े हैं