For your first question- the children/trustee would be listed as primary beneficiary. Your spouse is a co-owner, if/when one of you dies the other owns it 100% at that time. The primary beneficiary would come into play only when both are deceased (this could be important if you passed away at the same time like in a car accident).
On the second question I believe you only need to list the ss# for the beneficiaries themselves, not the trustee. For the trustee you probably just need legal name, address, phone number etc. The ss#'s are likely needed for reporting requirements for the IRS, the Trustee would not own the asset themselves in their own name so their ss# is irrelevant.