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I am sorry to hear you are worried about Tiger. My name is Rebecca. I am a veterinarian and will do my best to help.
Lethargy and vomiting can be signs of many different things in older cats; some are serious and some are not. Is he drinking more water than he used to do? Is he losing weight?
He could have kidney disease, hyperthyroidism, inflammatory bowel disease, so many possibilities. Normally I would recommend starting with a good physical exam and some blood tests but it sounds like that is not possible for you.
The hair loss may not be too significant. If he has any matted hair, from not grooming as well as he did when he was younger and more flexible, if the mat peels off you see a strip of hair loss like you describe.
If possible, I would have someone take him in for blood tests. If that is not possible, I would just stay in touch with your petsitter while you are gone.
Let me know what other questions I can help with, or answer.
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Let me know what other questions you have and how else I can help.