On your name change question, the answer is a definite "No." Your account records are required to reflect the true ownership of the account. A trust's name does not change just because for whatever reason the trust changes from revocable to irrevocable. If the trust is to continue after an event (for example, the death of the last of the grantors where a revocable trust does not provide for a new trust or other mechanism that would change the trust's name but the trust becomes irrevocable simply because it provided only that the grantor(s) had the right to revoke it), the new trustee(s) may take steps to restate the trust or otherwise change its name to reflect a change from "revocable" to "irrevocable."
So, if your bank holds an account in the name of the "Smith Family Revocable Trust" and the last of the grantors dies, you may note in your records that the grantors have passed and the trust is now irrevocable, but you don't change the title unless you receive documentation that the legal work has been completed to change the name of the trust.
It is true, however, that the trust can no longer use a Social Security number of a grantor; it must obtain an EIN even if the trust will be dissolved before the end of the year (because the SSN of a deceased individual cannot be used), and your bank should require that the new trustees take care of that.
If the revocable trust already has an EIN and it changes its name, it should check IRS requirements to determine whether it needs to obtain a new one.