Euryale

IMO, yes. Eury has weak stats and is only used for her modifier on NP which needs to be boosted by Tama's and her own Arts performance boost to be able to easily one shot Gawain or in the least put devastating damage in my experience also it makes it that her already extremely good NP gain would be better enabling her to potentially spam her NP with an arts brave chain.
It's best to be able to make all her skills to level 6 for the cooldown in the least, and it's best that her NP is at least NP5 for best results.

Well she's quite good Asset for long term gameplay. Like mentioned above at least lvl her skill to 6. Not just used in Camelot, but a lot of fight with males too.

It's really more of a 'win better' condition. Euryale's primary usage is for stunlocking male bosses. You use her charm skill and her NP to constantly stun and keep the enemy from firing their NP. But if you want to use her to actually kill bosses, then yeah, you want her arts up skill and her NP interlude. These are important.

For skills, they're nice, but chance-based with high cooldowns and therefore don't matter too much if you don't plan on using Euryale in the long term, but level 6 is always a good goal, if for nothing else but shorter cooldowns.

The short answer: if your goal is to use her as your boss killer, yes, you need her third skill, and at level 6 at least.

Yes, arts up to level 6, or even more. Not only makes her NP damage strong, the NP refund from arts card is nice as hell.

As for the others, Record Holder/Love Potion CEs can replace the level 6 charm, still put it on level 4 at least. Otherwise (using Formalcraft, MLB Kaleido Sapphire...) do level to 6.
Vampirism is chanced even on 10, yet it drain more on higher levels, put it on 6.