If your new rig will boot with your old windows, you should be able to install the new drivers and be OK.
That is not a bad idea if reinstalling lots of apps on a clean install will be a big problem.
As to activation, you will not have an issue if your windows 7 copy is retail or you are running windows 10.
Even then, you are likely to be able to activate with windows 7 OEM one way or the other.