In which calendar month of every year, each one of the first four dates (i.e., from 1st to 4th) is observed as a World/International Day for a specific purpose?

The correct option is October.

Let us verify this by listing the specific World/International Days observed on the first four days of October:
1. October 1st is observed as the International Day of Older Persons (and also World Vegetarian Day).
2. October 2nd is observed as the International Day of Non-Violence (marking the birth anniversary of Mahatma Gandhi).
3. October 3rd is observed as World Temperance Day (and traditionally World Habitat Day is often observed on the first Monday of October, which frequently falls in these first few days).
4. October 4th is observed as World Animal Welfare Day.

None of the other months listed (June, July, or September) have official World/International Days designated for all of their first four dates consecutively. Therefore, October is the correct calendar month.