How does mana cost reduction interact with colored mana in a cost?
Mana cost changes affect specific elements of a mana cost only. If a reduction says “Spells cost {1} less to cast” this will only affect generic portions of a mana cost. If a reduction says “Spells cost {U} less to cast” it can make spells cost {U} less or a corresponding amount of generic mana.

A good illustrative example is Khalni Hydra, which has a mana cost reduction specifying colored mana symbols (in this case green). This mana reduction can pay for generic mana costs (in the event an additional mana cost is added by another effect). However, the same is not true for generic cost reductions paying for colored mana symbols. This behaviour functions in exactly the same way as generally paying for spells, you can pay generic mana costs with any color of mana (or colorless) but you cannot pay color costs with colorless mana.

To summarise, Colored mana costs can only be reduced by effects that specify they will reduce that color of mana. Generic costs can be reduced by colored mana cost reductions OR colorless ones. All of this functions just like spending mana to cast spells.

To answer your specific examples:


Nemesis of mortals will always cost at least 2 green mana.
Lazav's mana cost will be unchanged. The mana cost reduction is colorless only, and he has no colorless mana in his casting cost.
You could discount 2 from the generic cost, but you would still have to pay the green portion of the mana cost. If you have 2 creatures in the graveyard and Animar with 2 counters, you will get a total mana cost reduction of 4 colorless.