Fairies known as fae, are not something that you can possesse. Its a spirtual being why would you want to have one. Try a summoning spell to become friends with them and offer sweeten milk and hunny or any sweet treats. They won't come at first but eventually they will learn to trust you and they will offer a helping hand but they will trick you but its all for fun, but remember nothing will happen if you don't belive. It will help if you ask Pan for help
They are free willed. Not all are good. They represent nature. There are good and bad sides of nature. Thus there are good and bad fae (faeries fairies). They are spiritual beings. Befriend them. They will talk to you. They don't like being summoned. Call them. That's giving them a choice to come. Not tearing them away from where the love.
One point to add surtr, there are different fae for different elements, the most common people meet are earth faeries, as they live inland like in forests, meadows and in gardens.
Well you cannot capture one in a bottle, or have one follow you around. Like many spiritual creatures, you work with them, ask them for help, stuff like that. Faeries aren't sweet little things, they're more like TinkerBell from Peter Pan, they are kind and loyal to those they care about, but spiteful and dangerous to those they dislike. Faeries are said to be everywhere, observing Yax occasionally helping or hurting those they deem worthy. They love generosity but loath greed. If you have a fruit baring tree for example, leave some fruit behind. If fruit falls to the ground, leave it, taking everything away can upset a fae.
I don't know about your wand because you simply made a statement. Without information on your wand, no one can really help. It could be the wood, how you made it, how you blessed and charged it. Could be a number of things.