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.
The difference between face value of a bond and lower market price, attributable to rising interest rates. On a sale, gain on the bond is generally taxed as ordinary income to the extent of the discount.