As you correctly note, a special needs trust must obtain its own EIN and is a separate taxable enntity. The threshold for a trust to file an annual income tax return on Form 1041 is $600.00 of gross income (before deductions).
Of the sources you describe, the gifts from individuals are clearly not income. I am not 100% certain about the money from the state. The absence of a Form 1099 is not necessarily determinative. There are exemptions for certain disability payments, however, and you should have your tax advisor check those.