Yes if lux catches you with a q you will take a lot of damage.
The difference between lux and zoe is that fact that lux has to actually make a decision if she wants to use her e for waveclear, save it for poke or for the one shot. Lux has to debate her decisions based on cooldown.
Zoe on the other hand has little to no cooldowns, and we will very soon see the same exact problem that taliyah had when she was released. Spammable, low mana cost, low cooldown, aoe abilities that clear waves starting from level 1 breaks the game because of how much stupid pressure it creates. Taliyah could q the level 1 wave and shove in, get vision in the enemy jg, and control the game from there. There is little difference from this problem as there is with zoe right now.
It's not because the damage, the damage is absurd but its not why she needs fixing immediately before it goes ANYWHERE near the competitive scene. The fact that zoe, compared to lux, has to make little to no decision making based on her cooldowns is ridiculous. She takes little skill compared to other mages because she can spam q, spam e, pick up ws, spam rs whenever she wants and theres no way to punish her because she'll q you again in 1.5 seconds. She'll shoot another e in 3 seconds. She'll r again in 4 seconds. Oh look she picked up glp or protobelt, welp I guess she just shoves me in and harrasses my jungler and theres nothing I can do about it.
Her waveclear needs to be addressed, and her cooldowns need to be addressed.