I remember reading somewhere a few years ago concerning one of the companion mods, Constance or Julan, and that it is good practice not to move them from their relative load order position when adding/removing other mods during a current game, as this possibly make their scripts mess up in the save.

Is this correct and if so does it also apply to non-companion mods that rely on scripts like GCD or NoM?

The general consensus was that moving the position of a companion mod was not a good idea. You could add mods that load after the companion, but not before. That doesn't apply to script heavy mods that I know of because it's not the scripts that cause problems, but NPCs that have scripts attached to them. I've always advised people to load stuff after my companions.

Okay, thanks Jac. I'm going to be possibly shuffling my load order a bit mid game and I wanted to be sure. This game so far has been crash free so hopefully it stays that way.

Follow up question. Is adding and removing esms considered changing the load order of esps as well? I have my companion mods at the top of my esp load order but I know esms load first. So if I remove an esm. are my companion mods in danger of experiencing the negative aspects of having bumped them?

That I'm not sure about. A safe bet would be to start your game, have your companion(s) wander around (not stand in one spot), make a new save, do the adjusting you're planning on, load the new save and have them rejoin you. If nothing is amiss, you're golden. If something does go wrong, just revert back to the old load order and try again. I've never had any post any issues with Jasmine so the whole not moving stuff around may have only applied to older companions that held most of their variables in their local scripts.

May be. Better try some test saves as Jac suggested.
Anything loading before modx (being .esp or .esm) could cause problems to modx and below in savegame if removed (although this is more rare nowadays thanks to Morrowind Code Patch fix which may fix references, but not for scripts I think).
Mods using targeted global scripts can still lose scripts target if load order changes, and also mods referencing persistent statics/activators from scripts can lose them.
If testing shows that problems arise, a way to avoid problems could be instead of removing the .esm, replacing it with a dummy empty .esm (Mash provides several dummy .esp usable as placeholders in Mopy\Extras\ folder, just copy to .esm one of these and redate it) having the same timestamp so loading order is preserved.
[EDIT]loose->lose, trying to make more sense

The idea of using a dummy esm/esp is excellent, thank you.

Of course that still leaves the problem of adding an esm. I think I'll just avoid doing that altogether. I had problems with both Constance and your Guars mod last time I played and I have a strong suspicion it was due to adding/removing mods. At least I know I can now remove mods without risking it.