Everyone likes food, yet not everyone enjoys the same particular food. It has the word yet in the sentence. If you remove yet, then there is two complete sentences present.
Temporary and con and that will give you the meaning
1) How do you deal with having to take care of your self and also take care of your pets well doing? 2) How do you feel about pets and how they can help with your mental health? 3) Do you see pets as a good way to cope with mental health problems? 4) Do you think you should have pets if your mental health is not good or do you see it as a way of coping and learning?