We also like to stay home and work on Duolingo!
Is the "zuo" really necessary for this to be correct grammatically?
It sounds unnatural, or is at least not idiomatic, without it.
Yesterday, my Chinese teacher corrected my writing and said I should never say "比较老的女人". Apparently, it sounds very offensive, so is it OK to say "老人"?
Yes, you shouldn't ever call a woman old! But yes, you can say 老太太 or 夫人 as a sign of respect for people older than you.