No. Gifts are not taxed to the recipient, regardless of amount or whether they are from people in or outside the United States. If you make gifts, you need to check on whether you must pay any gift tax (not income tax) and/or whether you have to file a gift tax return, Form 709.
A trust created during the lifetime of the person who created the trust. If irrevocable, income on the trust principal is generally shifted to the trust beneficiaries.