Can I use an Apple gift card toward the purchase of a new Apple Watch?

Not sure what happened with your purchase, but I have had no issues using an Apple Gift Card and paying the balance with a credit card.

Only if the price of the purchase does not exceed the amount in your Account balance. Apple does not let you use your account balance in conjunction with another payment method which they do not disclose until it's too late.

Yes.

I just did that last Wednesday, paid the last $38 from a credit card after using my gift cards.