Do not try to contect them if they see no intrest in contecting you. Trouble is what will come your way if you do. Be happy with what you know. So do not look for them just to pove they are real.
It is true that fae are not all friendly but as long as you mean well in your magick and your intentions they will not harm you. If you offend them in any way they will either disappear and never come back or become a big nuisance for you.
It is also true that they take a particular interest in certain people but they can be open minded to anyone who cares for the earth (e.g. If you dont litter, recycle etc).
I would also like to contact or summon/trap members of the lesser fae. No sidhe lords or anything like that. I would like to trap a sprite or a gnome or something.
Personally I find that extremely disrespectful, Fae are the protectors of our earth trapping them would benefit you how? Somehow I dont think many fae would be stupid enough to get trapped either that or it would be an extremely trusting fae and you tricking it would be very sad. Also I'm sure the fae's friends wouldnt be very happy.
Now you were saying you wanted to summon one, Im not sure if I want to tell you how to since you have the intention of capturing one of there magnificiant beings :( I hope you come to realise that the fae are not to be messed around with.
Not only are they extremely cautious of humans, because there are people like you who would do something like that, they are sensitive to emotion/intent. You have the intent to trap, or harm, and they pick up on that.
With greedy and malicious intentions such as these, the Fae will never get near you, or anything you produce.
Should you somehow manage to trap one, though trust me in the fact that you will not, they and most likely their brethren would exact revenge. Though the Fae are generally peaceful, they do not forget a debt- whether that debt is one of gratitude or vendetta.
Some of the earliest stories say that hawthorn acts as a mediator between this world and the world of the fae. While, yes, they can be found all through nature, some believe that it is easier to perceive them around hawthorn.
One story talks about how those who fall asleep beneath a hawthorn bush wake to find their selves in the realm of the fae.
While those are just stories, it is wise to take them into consideration. Many of the old magical ways are somehow preserved in legends and stories. At the same time, you have to discern the truth from what is created purely by imagination. Try looking into what the older stories of fae say; the ones that come from Ireland, Germany, and the middle east. As far as I know, those are the regions with the oldest recorded stories of fae. Then again, I'm no expert.
You need to be careful around the fae. Don't get them angry in any way. No matter how benevolent they are, from the things I've heard, they can have quite the temper. And if you try TRAPPING one... Damn. To be honest, I don't know WHAT they would do to you, but it would NOT be good. They got pissed off at my mom when she cussed them out after they stole her keys (there was a fire in the neighborhood and firefighters were evacuating everyone and she needed keys to drive). They threw the keys right at her face and never came back. And she had good cause to be cross with them. YOU on the other hand have no reason to capture them. So they most likely will be much harsher with you, I can imagine.