This post makes me very angry ok... I hope you are not thinking of a make believe fairy. This is disrespectful to the fae they are wonderful beings they are not here to be someones pet treat them with a bit of respect please. They are not all friendly some fae can be quite nasty and a fae would never want to be a pet.
Now as for the whole summoning thing some will argue its rude to summon and others will argue its not but my opinion is to just respectfully and kindly ask them to be present with you but in your case they would never come to you to be your pet, Im sorry but your wish cannot be granted.
I think of fae as equal to me, to all of us. They are beautiful people who tend to the world we are destroying why would they help us after what the human race has done? Because they are kind and not all humans are like that. The kinder fae usually happily help someone with their magick but there should be good reason behind the magick and they should especially help people who are kind and selfless themselves. So I my hope is that you now understand where you have gone wrong.