"I know that only one transfer per year is allowed,"

Think of it as "direct or indirect."

Transfers between the trustees, or direct, are not counted.

But every time she took the money, "touched it" first, that means it was an indirect = went through her hands, first.

And you stated the deposit for the third event was just as a brokerage or savings account, then. So, the answer to Q1 = just regular banking. Not a distribution, but a withdrawal. If there wasn't enough of a cash balance in a brokerage account, then something gets sold to accommodate the withdrawal, and there will be a 1099-B.

Question 2 answer: none of a Rollover is considered to be Contribution. These are two different ways money goes into IRA accounts. If there was not also more money put in, for the second deposit, or put into the initial account in the first place, you have no contribution in what you described.

But you also might be overlooking, if this is for 2020, there is a specific provision for "take it out, then put it back" that is Not counted against the once-a-year provision, due to covid as disaster.