Let's say I posted a question to which I received a good answer, and I accepted that answer. Now, I wish to add something more to the question, and add a bounty to the question. Is this answer eligible for the bounty even if nothing is added to it during the bounty period? If the other newly added answers prove to be unsatisfactory to me, could I award the bounty to this answer?
Yes, you can do it. The pre-existing answers are not eligible for automatic award of bounty, when the bounty setter fails to award it themselves. But if you do award the bounty, there is no problem giving it to an old answer. In fact, one of the canned reasons for starting a bounty is "reward an existing answer".
For the complete description of the rules, see How does the bounty system work?