Do I have a good CFI?
I'm a CFI/MEI.  I have 700-800 hours teaching private pilot students in the part 61 environment.  That's not an awesome amount but I think it's enough to comment on this question.

This guy is wildly unprofessional, maybe even criminally negligent.  You are paying him to observe you, to provide feedback, and to keep you safe.  He can do none of those things while he is sleeping on the job.

You have a lazy instructor and I'm curious to know in what other ways he is failing to teach you.

It's his job to excercise good judgement and scrub a flight if he's not up for it.  If he's overworked he shouldn't be flying.  That's his problem, not yours.  He has no excuse for his behavior and he's setting a terrible example for you to follow.  DO NOT be like your instructor.  I hope you're not using him as a role model for what a good pilot should be.

He's failing to demonstrate sound judgement and showing you that complacency is OK.
He's not challenging you or looking for teachable moments.
He's exhibiting classic hazardous attitudes...invulnerability and anti-authourity at a minimum.

I'd talk to him first and then fire him if this nonsense continues.  I'd also demand a refund on CFI billing for his nap time.

And, yes, there should be a defined lesson plan with expectations for each lesson.  This goes for pt61 and pt141 schools.  Every CFI should have created plans during their CFI training.  I have no idea how a CFI applicant can pass a CFI practical or oral exam without a comprehensive collection of lesson plans.

No, it's not OK for you to be in charge of what is covered.  You should know what the next lesson will cover so you can prepare.  He needs to give you the training plan and he needs to drive it.  Again, it's his job, it's why you pay him.

I think this guy is milking you for billable hours.  You might just be a cash pinata for him.  Either that or he's just incompetent.