T-bills have regular auction times (barring holidays)

4/8/13/17/26-week T-bills are auctioned every week. The announcement date for 4/8/17-week is usually Tuesday and Thursday for 13/26-week. The auction date for 17-week is Wednesday, Thursday for 4/8-week, and Monday for 13/26-week. The settlement date is a week after the announcement date (usually).

Answer to your questions:

2. Have the money ready on the announcement date.

3. On the announcement date. (I think it's easier to do it on the announcement date)

4. Use auto-roll if you want to keep rebuying that specific term length.