As Lyonaria said, do not chuck any post. In 2.5 years you'll thank yourself as you won't be running around like crazy (like me) trying to find every bit of post possible. Things like NHS letters, utility, council tax, work letters etc are great ones to keep. But keep them all anyway as there is a requirement for 'different sources' so filter through what you need when the time comes.
Regarding the second part of your post, you're now in an unfortunate loop that everyone finds themselves in. You need proof of address to get into the doctors, but you need proof of address just to get proof of address. My wife is with Lloyds Bank and they started an account for her with no proof of address. I highly advise trying them first to at least get the proof of address cycle in motion. Then other things will fall into place.
Book yourself in to get your NI number as it's needed to work, as I'm sure you know. Be vigilant with your doctors, you have a right to the NHS. This is a warning just in case they argue with you, as most aren't aware of the visa allowances. However, my doctors didn't batter an eyelid with my wife.
Lastly, your child will be a British citizen once born (as well as American) so will not count towards your second visa application (FLR-M) in terms of minimum earning requirements etc but will need mentioning.