When using essential oils topically, it is best to dilute in a carrier oil. A carrier oil could be coconut oil, jojoba oil, sweet almond oil, or the oil of your choice.
Essential oils can be toxic to cats. Their liver lacks a detoxifying enzyme. It is best to avoid the oils around your feline friends.
One common oil that should not be used on dogs is tea tree oil. There are many other unsafe oils such as wintergreen, pennyroyal, and the list goes on. A hydrosol would be a much safer option or consult and aromatherapist for your canine friend.
Generally when using a diffuser it is best to run it intermittently. It is also important to consider who else is in the room when diffusing. Some oils can trigger asthma attacks and not all oils are safe for small children or animals.