Is this fine to do?
For the continental / lower 48 states of the U.S. this shouldn't be a problem, however you might need to isolate Hawaii and Alaska to treat those states separately. The projection you are describing is a composite projection, meaning it's actually using 3 different projections to fit the lower 48 states, Alaska, and Hawaii in the same map area.
If you're using D3JS, the d3-geo part of the library has a good explanation of the map projection you are using.
All three areas use the Albers’ conic equal-area projection, and where you "center" that projection is up to the map maker to decide. The U.S. Albers is one set of parameters optimized for showing the lower 48 states of the U.S., so when your application "zooms out" to the whole map area I would recommend re-setting the projection parameters, as it will likely look odd to have the lower 48 states rotated. That being said, it's up to the map maker to decide.
And how is the same rotation viewed if done within the US projection as well when zoomed into a state that you can still view surrounding states?
A related piece of prior-art is Noah Veltman's "State Plane Shuffle" which gives you a sense of how the U.S. looks when the map projection is centered on a particular state (or part of a state), but the rest of the states are included on the map. My opinion is that when you are "zoomed in" or cropping the map area to a particular state, it's appropriate to show neighboring state boundaries, but when "zoomed out" to fit the lower 48 states, you would likely want to revert to the U.S. Albers projection settings.