What is the difference between cuarto and habitación and dormitorio ?Feel free to just provide example sentences.

They are synonyms. The three of them mean the same. If you want to be very picky with the words, then here´s a deeper explanation.
A "cuarto" is any room at the house, it could be a bathroom, your personal room, the study. The exact meaning of "cuarto" is room.
Habitación is where someone stays, it could be "your parents room", " your sister´s" or even "your´s" I suggest you use it for the matter of a personal room only.
Dormitorio is a small room where there´s barely even space for a bed and a desk, the purpose of this room is as its name indicates "to sleep" para dormir.

Here are some sentences:
Cuarto "En este cuarto guardamos las cosas de mi papá. Ocupan mucho espacio". "Este es el cuarto de juegos, tenemos una mesa de billar" "Rayos! Dejé mis zapatos en el cuarto de mi primo"
Habitación. "Esta habitación es muy amplia!" "Siempre tengo desordenada mi habitación" "Vamos a pintar la habitación de Juan de azul aun que el no quiera!" "Esta era la habitación de mi hermana, pero ahora es el cuarto de manualidades de mamá" (To be honest, habitación is used but we are more used to cuarto)
Dormitorio (we rarely name our rooms "dormitorio", this is more used when you are at some place that is not a house, like a university campus, a hospital, a summer camp, you know, places that have an area for sleeping. That's what we called a "dormitorio" the room or area for sleeping.
"Las camas de los dormitorios en este internado huelen tan mal" "Dicen que lo más horrible son los dormitorios del hospital" "John, compartirás el dormitorio con Alan, tu nuevo mejor amigo"

Sorry for the long reply, hope it helped :)