You should be able to find the answers to most of your questions in your Proprietary Lease. In the PL for my co-op, paragraph 16(b) states

"16(b) If the Lessee shall die, consent shall not be unreasonably withheld or delayed to an assignment of the lease and shares to a financially responsible member of the Lessee’s family (other than the Lessee’s spouse as to whom no consent is required)."

The key here w/r/t your daughter is the term "unreasonably withheld". Under most circumstances when a person dies, the family member inheriting the co-op is an adult with a sufficient and stable source of income and/or assets. The reason your daughter will probably need to submit a financial package is to make sure she will be able to afford the co-op's maintenance and mortgage payments if there are any. The board wants to avoid the complicated and expensive eviction process if your daughter is chronically delinquent with her maintenance payments.

As PC#1 stated, the best way to make sure there are no succession issues and to get your daughter's name on the stock certificate is an informal conversation with the board president. He'll most likely tell you what could interfere with adding your daughter. If there are any issues, resolving them will make it a quick and simple process.

In a situation like this where the person you want to add is not your spouse (and I am very sorry for your loss) a consult with an attorney who specializes in co-op law will be money well spent.