Cure Wounds/Healing Word heal the living and do not affect the undead
Cure Wounds states:
A creature you touch regains a number of hit points equal to 1d8 +
your spellcasting ability modifier. This spell has no effect on undead
Healing Word states:
A creature of your choice that you can see within range regains hit
points equal to 1d4 + your spellcasting ability modifier. This spell
has no effect on undead or constructs.
Unless I missed something from the homebrew race you shared, you are indeed undead and can cast your bard spells normally.
Being undead does not change how the spells work, but it changes its effect on you, of course, since you are undead.
I was surprised to see that Inflict Wounds does not heal the undead in 5e. (Necrotic damage does not heal them, some are immune or resistant to it so... it hurts them)
On how to heal yourself as an undead creature, the question How can I heal undead creatures? can help you.
Yes, your bard spells will heal your living comrades, but you will have to find other ways to heal yourself since they don't heal you.