These are all questions of state law, but generally the answers to your questions are “yes,” “yes,” and “no.”  A current gift or a bequest made to a charity for its “general fund,” without further restriction, would normally be considered “unrestricted” and would be usable by the organization as it sees fit for its general charitable purposes.  You could use it to make grants and to pay for your operations, since the cost of operations is necessary for you to carry out your general charitable purposes.  You could spend the bequests all at once, use them to fill an operating deficit, or hold them as a “quasi-endowment” from which you spend income only.  Your board decides how to spend the gifts.  In most cases, the donor would not have standing to challenge the board’s decisions if the donor disagreed.  The donor normally gives up any right to control a donation after it is made, unless the donor specifically retains a right to enforce a specific restriction.

If the donor left the gift to a separate named fund created by the donor, and it was your practice to tell grantees from which fund a grant was made, you might want to follow the practice and notify the grantees who made the original gift to the foundation.  But a gift for a general fund is fungible with all the other gifts to the fund and allocating any expenditure to a specific individual’s gift would be arbitrary and impractical.

The gifts you describe for the “general fund” sound much more like a gift to the annual fund of a college or university.  My gift to annual giving (although undoubtedly less than your bequests) is unrestricted and I don’t suspect that when it is used for the president’s salary his direct deposit memo will note that part of his pay this month comes from Don Kramer.  The University thanks me promptly for my generous contribution every year but never tells me how my gift was spent, other than telling me frequently what it tells all alumni — that unrestricted annual giving is used to make the University an even greater place.