Cu Chulainn vs Elisabeth Bathory (Halloween)

I'm assuming you mean caster Cu? He is story locked, so unless you roll Fuyuki rate-up or the story gacha, it's unlikely you will get him to NP5, and if not, his NP damage is low compared to Halloween Liz. To his credit, he has one of the best defensive skills in the game, a 3-hit dodge, so he tends to be more durable. He also will soon get an NP charge skill, and he has 2 crit damage buffs, but they aren't that great on him.

They both really only have 1 real purpose that they are good for and that is farming assassins, and Liz is much better at that simply because her NP damage is higher and she also has a buster buff.

I'd definitely say so. He's tougher to NP5 then EliCaster due to being story locked, and his damage is way less consistent due to his generally awkward role as a critical damage caster. I think he's outclassed in pretty much every category.

Yes she is. I have her max leveled and she's my primary caster until I got Illyasviel. She has a Buster Np and Buster Buff, a Heal which gives you stars as well and guts. To give you an example, I soloed the Hassan Parade last Nero Fest just with Her with the event CE. No need to use Iri since she has her own guts and heals.