Fixed income - held to maturity

Under held to maturity method, if bond is buying at par, can we say the carrying value is always the same as the amount we invest? Even market interest rate up or down afterwards (different with coupon rate)

Yes. Held-to-maturity securities are not marked to market.

If you buy a bond at par, is there a premium or a discount to amortize? HTM securities are recorded on the balance sheet at amortized cost.