I used to be a nanny, am now a mum
showering at work unless there is a rare exception or previous agreement or overnight, not ok, but not a sackable offence. As a mum if my nanny overslept and the choice was come in late or come in and shower once there I would probably choose the latter. As a daily thing no, and I'd be having a conversation about it. As for supervision, is your child at risk when you are showering? It isn't a supervision issue for me it is taking the piss thing... ditto taking naps unless you know... e.g. nanny comes in with bad cold and says "I'm here but I'm sick, I will need to lie down this afternoon".
Using food. I would expect a nanny to eat at any time the child she was looking after was eating or would normally eat (if child is sick). I worked as a nanny for ten years and it was standard at every job no matter what hours I worked. If you don't want to pay for food it should have been specified in the interview and you should pay more. I would want my nanny to eat what my child was eating, 1) nanny is more likely to cook well 2) it is weird for the nanny to always eat something different and is confusing for a child. My employers usually had a tub in the fridge for special off limits things or a separate fridge in the garage or pantry for their food. Most of my employers left money for me to buy other food if I wanted it... e.g. french bread to go with the spaghetti and meatballs I was making for supper. Frankly it is unlikely I would take a job without food unless they paid much higher... it is kind of weird.
Using toiletries "please don't use our toiletries other than hand soap, if you would like to leave some of your own for your own use I will provide a basket for you to store them in."
being late, it happens on occasion... maybe once or twice a year tops. If it is habitual and not corrected then that would be a reason to let a nanny go IMO
sleeping in your bed... hmm hard one. If kid is sick and/or having nightmares or trouble settling it might be necessary and I'd never expect a nanny to share a single bed with a seven year old. How horribly uncomfortable. Otherwise not acceptable. Perhaps provide a large blow up mattress in case of this? Nanny is unlikely to use that unless she is desperate.
Medical appts... I took kids to medical appointments but it was agreed on before hand if not an emergency. If she works two jobs then she may be limited when she can go. Again a discussion "Please let us know if you have an appointment you need to go to so we can arrange cover if we don't want "sally" to go with you"
Calling in a sickie. Yeah, I'd be pissed about that big time. Really pissed. However, if she is throwing up and has a bad head do you really want her with your kid? I would make it clear though if she did it again we would be done. I would also worry that if she was that off her head she may still have alcohol in her system the next day while looking after my child (I really hope she isn't drinking to excess the night before then driving your DC).