As for alignment, well at 320x200 (or 240 if your video card+monitor could manage):
- the crosshair might have been on the HUD layer which was pixel-aligned.
- 2px wide would have been quite wide, and 1px would have been off centre.
Tradeoffs tradeoffs.
I remember thinking that Doom had this "always looking down the sights" feel and Quake inherited that but the 3D aspect of it was more like the weapon felt belly height instead of shoulder height.
Since weapon was centered it wasn't too hard to aim horizontally, but Quake having 3d look up down and LOS hitscan line (vs Doom which hit the first enemy along a vertical plane) made Q hard-ish to aim vertically.
Off-centre of your monitor, but not necessarily off-centre as far as the game is concerned: it would be trivial to shift everything one (or half?) a pixel to the upper left. Ie so that your shooting direction perfectly aligns with the crosshair. You could do that for any crosshair position.
Even cross-hairs that are off-screen.
† such as https://www.youtube.com/watch?v=4gNYTqn3qRc https://www.youtube.com/watch?v=zfY96tAqmdI