Only one taker for the Hebrew calendar quiz?
Well, if you want to work on it yourself don't read past this point. Spoilers follow.
Start with the axioms, as the mathematicians call them. Each month in the Hebrew calendar is either 29 or 30 days long, usually alternating between them, since the lunar month averages about 29.5 days. The first day of each month is Rosh Chodesh, and the 30th day of each 30-day month is also Rosh Chodesh for the following month. So a 29-day month has one day of Rosh Chodesh, while a 30-day month has two.
Consequently, between the end of one Rosh Chodesh and the start of the next, there are always exactly 28 days: days 2 through 29 of each month. That's four whole weeks. As a result, the next Rosh Chodesh always starts on the succeeding day of the week from the end of the previous Rosh Chodesh.
If Rosh Chodesh for month m ends on Tuesday, Rosh Chodesh for month m+1 must start on Wednesday. It may or may not extend to Thursday, depending on the length of month m.
How many days of Rosh Chodesh are there in a year? Twelve months, half of which have 30 days, should yield an average of 18 days of Rosh Chodesh. A couple of days either way don't affect the answer, though, since whether there are 15 days or 21 days, the number is more than two weeks' worth and no more than three weeks.
No matter what day you start, 18 successive days of the week (or 15 or 21) must include at least two Shabbatot, and no more than three. That's the answer to the first question: Either two or three.
The precise answer for a given year will depend primarily on which day Rosh Hashana starts. If it starts on Shabbat, Rosh Chodesh Heshvan starts on Sunday and the remaining days of Rosh Chodesh for the year will include only two Shabbatot (note that Rosh Hashana itself is not generally considered to be Rosh Chodesh, even though it technically is). If R"H starts on Friday, Heshvan will start on Shabbat, and three Shabbatot will fall on Rosh Chodesh that year. And so on.
The bonus question is indeed more difficult, and I'll leave it open for now. Partly to give you something to think about over Shavuot, partly because I haven't worked it out completely myself yet.