29 Iyyar 5768 (but if it's after sunset, it becomes the next day: 30 Iyyar).
Before sunset, it was 29 Sivan, 5698. (After sunset, the new day begins, so it would be 30 Sivan)
9 Tammuz 5756
Nothing. We go by the Hebrew calendar.
Av is the 5th Month of the year on the Hebrew calendar. It usually occurs in June or July
Your Hebrew calendar with have the times.
The Bengali calendar date equivalent of the 30th of June, 1935 is 16 Ashar 1342.
The Yiddish translation of the name June is יוני (Yoni).
It depends on how you count the Hebrew months, e.g. do you start counting from Tishrei or from Nissan. The third month of the calendar if you start counting from Tishrei is Kislev, which occurs around November-December. The third month if you start counting from Nissan is Sivan, which occurs around May-June.
A calendar for June 1947 can be found below this answer.
You can get calendar for June 1987 at calendarlabs.com
The name June has no meaning in Hebrew. Your only option is to write it phonetically as גיון The Hebrew word for the month of June is yooni (יוני) but this word is not used as a name in Israel.
A calendar for June 1944 can be found in the external link beneath this question.
See the related link for a full calendar of June 1946.