My advice for facilitating teamwork ( which you are already doing) is to listen to the cna's and then go and assess the resident as soon as you can. You will decide to give the med or not. After you have been there for awhile you will get to know your residents, their triggers and which interventions are effective for each particular resident.
You certainly can remind a cna that sitting 1 to 1 for a few minutes when you request she do so IS her job.Sometimes when you are a new nurse in LTC you 'll come across and aide or two who may try to test you.You will have to make your expectations clear from the start and follow through.If you tell someone to do something and they refuse to do it you'll have a problem if you permit it to continue.
As for the tylenol issue-I don't see a problem with any staff member asking a resident if they have pain. I don't believe that is out of the scope of practice. It's not different then asking if they are thirsty or hungry.The cna's know the residents best in LTC and sometimes the resident won't complain to the nurse.
I am a big believer in tylenol/motrin etc in the afternoon especially with the dementia residents.You'll soon learn to address basic needs first when a resident has increased agitiation (hunger,thirst,pain,toileting) That is often all you'll need to do. Their is a thread on here dealing with alzheimer's pateints -it's full of tips and tricks for dealing with behaviors.Do a search-it's a "not to be missed "thread. Good luck.