Thank you for your question for AAI.
This is actually a question we have received on several occasions, and I will share those links at the bottom of this answer.
The short answer is: it depends.
However, please keep in mind that Islam was revealed in many decades and no one should be expected to change over night. Please don’t feel like you should get rid of your dog right now, as if it was a priority.
Focus on what truly matters.
Focus on learning the basics of Islam. Focus on learning how to pray and on memorizing your prayers. Focus on trying to cut out seriously haraam things, like drinking, for example.
Specifically about Dogs
People argue about whether Muslims should own dogs, primarily because of this hadith from Prophet Muhammad about dogs:
Whoever acquires a dog, aside from a dog for hunting or shepherding, his reward will decrease every day by huge amounts. [Bukhari, Muslim]
The consensus among scholars is that dogs who serve a purpose in the home are welcome in the home. This purpose would include assisting those who are sight or hearing-impaired, or as guard dogs to protect the family.
So, if your dog is the sort of dog who doesn’t actually help in the home, most scholars would argue it’s really not ok to keep it. You mention your puppy is a shitzu, and that sort of dog is very small, and not really much able to help you.
The reality is that dogs’ skin and hair is different than ours, and is usually dirtier.
So we should be sure we are clean and good-smelling when we pray, and having touched any animal, even a cat, would naturally make our hands dirtier than they would be otherwise.
This is especially true if the dog licks you. Although you don’t need to make wudu, you should probably wash your hands.
If the dog has licked you, the best thing is to lightly rinse the place where he did lick you with water. Sheikh Faraz Rabbani says:
If a dog’s saliva comes onto one’s clothes, one must simply wash the affected area itself, until there is no readily-removable trace of the saliva left (such as by pouring water over the affected area three times, or placing it under a running tap). Doing so seven times or with dust (or other cleaning agent) isn’t a requirement but, rather, a recommendation. [ref: al-Halabi, Multaqa al-Abhur; Shurunbulali, Maraqi al-Falah; Ibn al-Humam/Marghinani, Fath al-Qadir `ala al-Hidaya]
Some schools of thought in Islam, like the Maliki school, argue that dog saliva, or really any animal’s saliva, is not unclean.
Regarding if Angels will come in to your house if there is a dog there, I would advise you to submit that question to our Ask The Scholar section.
The simple answer, though, is that Angels do not like to enter places which are filthy or smelly, and the cleanest dog in the world is still going to emit a powerful smell.
I hope this answers your question, and again, please don’t allow issues like this to overwhelm you. God is more concerned in your developing your spirit and drawing near to Him than these small things.