It can be uncomfortable to be in a new living situation with someone you don't know, especially if they're of a different gender. It's important to prioritize your safety and comfort, so if you feel uncomfortable staying in the apartment with the male roommate, it's okay to reach out to the pet owner and explain the situation. See if there's an alternative living arrangement or if they can make alternate arrangements for their roommate to be out of the apartment during your stay.
Regarding the cleanliness of the apartment and the lack of clean sheets and towels, you should also reach out to the pet owner and let them know the situation. It's important to have a clean and comfortable living environment while house sitting, and the pet owner should be responsible for providing that for you.
As for taking the dog to your apartment, it's generally not a good idea to take a pet out of the owner's home without their permission. Plus, you'd need to consider whether your apartment is pet-friendly and if you're able to provide the same level of care for the pet that the owner expects.
In summary, communicate your concerns with the pet owner and see if there are any alternate arrangements that can be made. Prioritize your safety and comfort, and try to work with the owner to ensure a clean and comfortable living environment while house sitting.