Michael Satlow has a publicly available list (digitized from Aaron Hyman, Toldot Tanaʾim ṿe-Amoraʾim : mesudar ʻa. p. a.b. ʻim beʾurim ṿe-hagahot ṿe-girsaʾot shonot) of Tannaim and Amoraim.
Yes, there were later Hillels and Shammais. He has (based on an Excel spreadsheet version I have privately) the following later people:
- Shammai (I) the Zaken
- Rabbi Shammai, a 4th / 5th generation Amora from Eretz Yisrael -- for this, see e.g. Yerushalmi Demai 5:1, and others listed by Hyman.
- Hillel (I) the Zaken
- Rav Hillel (I), a 4th / 5th generation Amora from Bavel
- (Rabbi) Hillel (II), a 1st generation Amora from Eretz Yisrael
- Rav Hillel (II), a 6th generation Amora from from Bavel
- Hillel (III), sometimes with the addition of "the brother of Gamliel Zuga).
There are also several Hillels who have patronymic or who incorporate place names (e.g. "from kifra"). But the aforementioned are sufficient that a clarifying HaZaken might be necessary.
That doesn't mean that that was the necessarily the reason for HaZaken. But in terms of point 3, there indeed were others.