We also had heard that greyhounds don't bark, but ours must have missed the memo...it sounds like yours may have, too!
My partner and I both have anxiety and she certainly tested our nerves a bit the first week or two. Although I grew up with dogs, she's much, much bigger than my dogs were, and her barks can be intimidating, especially when they seem to happen for no reason! The bowing and "smiling" intimidated me a bit at first as well, but that's just how they show they are happy and want to play -- now it makes me feel very loved when she shows her goofy smile!
I think you're on the right track with reward-based training. One thing I would try is putting his toys where he can get to them at all times -- that way if he is bored, he can amuse himself. If he barks, even though it's difficult, try to ignore him (unless he's obviously distressed or needs help). Don't "reward" him with any feedback, even if that feedback is saying "no". Instead, do whatever helps you to control you anxiety, whether its deep breathing or channeling a peaceful image -- and try to ignore him. Get up and do chores, pour yourself a glass of water, etc, as if he's not even barking. This has helped with ours, as she realizes barking doesn't get her any attention at all. Try to make sure you aren't locking him into rooms that you aren't in, as greyhounds generally like to accompany you instead of being alone.
Does he have a bed or other designated spot for naps? That might help to give him a place to lay calmly while you're on the couch or otherwise occupied. If you see him doing calm activities, like napping or even just being quiet, be sure to praise him and tell him he's a good dog, and reward that "normal" behavior too. We spent a couple of weeks giving out treats and praise every time ours laid on her bed, and it encouraged her to continue doing that. Eventually, we started giving treats for laying on the couch, and now she spends plenty of time sleeping there, too. Also, do you have a yard he can run in? If so, try some reward-based training outside in place of one of his walks.
I expect he will settle quite a bit over the next few weeks. The change your new dog is going through is huuuuge, but it's okay to feel frustrated and anxious about the change on your end, as well. The people on this forum have great advice, but sometimes everyone's passion for greyhounds being the best dog ever can make new owners feel a little anxious that they're doing something "wrong" when their hound isn't behaving perfectly. It sounds like you're doing a great job, he just needs a little more time!