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My name is ***** ***** I will be glad to help with your question.
I'm sorry Lily is experiencing these symptoms. There are various possibilities of what could be causing her mouth rubbing and lip licking. First, whether or not she is an indoor-only cat, she can still be bothered external parasites like fleas or mites; this may be itching her mouth/lip area and causing the rubbing. She also could have allergies to some ingredients in her food and/or contact allergies to something her mouth is coming in contact with, like her food/water bowls. The best type is glass, as it can be washed thoroughly and retain no bacteria.
The licking of her lips may be a sign that she's nauseated. If she is eating, drinking and eliminating as usual and not vomiting at all, she may not be nauseous.
I'm glad you were able to check if anything was stuck in her nose and ran your fingers along her gums; that was a very smart move. Her gums should be pearly pink and moist. I'm also glad you're about to take her temperature. Normal for cats is between 100.5 and 102.5 (rectally). If something is stuck up her nose, it would be difficult to see; the vet would most likely be able to see more and further; an x-ray may be necessary.
The fact that she is not spayed is not good for her health. This increases her chances of developing a serious uterine infection called pyometra and also increases her chances of developing mammary cancer. Please discuss this issue with your vet and after performing some lab work, the vet will let you know if she is a good candidate for spay surgery at this time.
She should be evaluated in person by her vet, in any case, to find the cause of her rubbing her paws over her mouth and licking her lips.
I hope all will be well with your Lily! Please let me know how she's doing and what the vet's findings are. Thanks!
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