While teenagers who are adopted may miss out on benefits offered by the social welfare system, I doubt that they feel that they're losing.
1. Very few children are adopted as teens. Those that are have the resources of their new families to draw on.
2. The resources of the social welfare system for ex-foster kids who have aged out of the system are *terrible*.
There isn't a general amount for scholarships. I've seen scholarships as small as $50, and as large as full tuition, plus room and board at a top tier private university, plus a personal stipend. How much is up to the organization granting them - and if you want to do this, I strongly suggest starting a registered non-profit. Otherwise, it's just you giving gifts to selected strangers, it may have gift tax implications (depending on how you do it), and you don't get a tax deduction.
Also, if the scholarships are granted by an independent entity, that entity can be given money or other assets in your Will, and can continue to fund scholarships after you die. This may seem like long-term thinking, but if it's just you, and you get hit by a tomato truck, some poor kid who was relying on your money may very well not get it.
Alternatively, as suggested, you could work with an existing organization. That would take a lot of work and worry off of your hands.
Different scholarships work different ways. Some scholarships are given as personal checks written directly to the student - the logic there is that money charged by the school isn't the only cost of an education. A scholarship can help a student buy books and supplies, cover living expenses (so that they can work less and have more time to focus on their schoolwork), get tutoring, or pursue educational opportunities like attending a conference in another city, or spending time in a foreign country learning the language.
Other scholarships are paid directly to the school, on behalf of the student.
If you do this, how it's done is up to you. However, please be aware that if you, as an individual, write a check to pay someone's tuition directly to the school, then you don't have to pay gift taxes on the transfer. If you write the check to the student, depending on how much you give, there may be gift tax implications.
In terms of selecting students: essays need not be about the student's life. They can be about the student's plans. They can be about the student's favorite television show. Either get some personal information (via interviews or essays or what you will) or pick names out of a hat. Up to you.