As I understand it, it is not technically that your visa lasts for 12 months, but that it converts to a temporary green card for 12 months. It makes no practical difference for travelling, but it is a practical difference for if you need proof of residence for anything.
As to above yes it is fine, as long as you have the old passport with the visa. The validity of the visa is not dependent on the validity of the passport. And anyway by your second entry as I described above it is no longer a visa but a temporary green card.
Also re taxes, yes your entry date marks the date you are liable to start filing, but the first actual filing will be the following calendar year.