You can close & fund your new purchase loan at the beginning of the month & have the option to pay or NOT to pay the remaining months interest. I believe you have up to five days into the new month where you have the luxury of this option. Ask your lender.
The answer is ... it depends. When you close, the settlement company will collect interest for the remainder of the month so, if you settle on the first of the month, they will collect interest for the entire month. However, since interest on mortgage payments is paid for the previous month, your first payment won't be due for 2 months. For example, if you settle on June 1, you will pay interest at closing from June 1 to June 30. But your first payment won't be due until August 1 because you'll be paying the interest for July. Of course, if you're paying cash, it doesn't make any difference at all.
Doesn't matter too much - at the first of the month, the buyers costs are higher (because you are prepaying interest for the duration of that month), at the end of the month it is less. The reverse is true for sellers. Other than that, it is driven by personal agendas.