Do I use my name and my social security number in order to proceed with the online application?

Use the name and SS# of the trustee.

Be careful. The trust will be identified to the IRS by its EIN (a.k.a. "taxpayer identification number.") On the application, there is a place to identify the responsible person, including that person's SSN. But the trust will be identified by its own TIN. The responsible party would usually be the settler (I think that's what you mean by "originator.") I gather you're not the settler. In 15 years of doing estate planning, including trusts, I have done a handful of irrevocable trusts. Usually it is neither necessary nor desireable. The fact that you're asking the question about the EIN makes me think that you are do-it-yourselfing an irrevocable trust. Have you thought through all of the tax implications? Are you ready to prepare and file a trust tax return each year? Are you really sure this is what is wanted? I would put the brakes on and get some professional advice. It will be much easier now than later on.

You cannot use the original grantor's social security number if he/she is deceased. Your number would not be associated personally with the trust. Make sure you follow the instructions for completing the form.

You would use the Trustor's SSN since that is who established the Trust. You must provide the SSN so that it cab be traced back to the Trustor.

Yes, you use it for the application. Once the EIN is issued, your SSN doesn't really matter any more.

In order to get an EIN for the trust you will need all of the numbers for the settlor as well as the trustee.

I advise you obtain some probate/trust administration advice from an attorney who can answer your questions or you can ask Social Security for answers to your question.

A revocable trust can use the same EIN number as the individual originator of the trust (generally his SS# is used).. An irrevocable trust needs an separate EIN number use the SS # of the originator of the trust (with his permission of course).