I've got an older laptop with Intel HD 4000 which I use for traveling. Will the whole Lords of the Realm series (1,2 & 3) runb on a HD 4000? Do these games run on Windwos 10 at all?

I can confirm that LOTR 2 runs on Win10.

Depends whether the version of LOTR1 on here uses DOSBox or is the later Windows version. The latter may have problems with Win 10
Either way, LOTR1 doesn't require anything from your graphics - it came out waaay before hardware graphics acceleration was a thing for games.

I have a thinkpad with Intel Graphics HD 3000 and the game's graphical performance is terrible. Everything is happening slowly, animations are slow, I see pixels with holes. I am running on Windows 10.