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So when creating/modifying monsters how much do things like immunity and resistance etc., affect CR?
This is a purely subjective, and are things the GM needs to consider in the overall balance of the monster. In general a monster's offensive capabilities are more impactful than defensive ones, and the only defensive abilities you should be mindful of are those that render a monster completely impervious to common tactics.
He's been trying to combine the various types of golems with the hopeful (though likely not entirely fulfilled) end goal of creating a golem with all of the benefits of multiple golem types without any of their weaknesses. The idea would be to combine say Clay, Iron and Stone golems to get their advantages (and any healing from certain damage types) without suffering the susceptibility to slow and the like.
Not at all; golem weaknesses are too specific and circumstantial for removing them to be worth an increase in CR. Since you're keeping all the other stats the same, the CR shouldn't change. CR is only a rough ballpark estimate in how challenging a monster is to defeat; depending on party composition or situation it could be much easier or harder. Obscure or situational weaknesses therefor should not factor in.
A good way to think about this is through CR equivalencies: two CR 14 monsters should be roughly as challenging as three CR 13 monsters, since both are CR 16 encounters. Does the removal of weaknesses make two Iron Golems as challenging as three standard Iron Golems with their weaknesses? In almost all cases the answer is obviously "no", the loss of one body drastically cuts the HP and damage output of this mob and renders them much easier to brute force regardless of any weaknesses.