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Yes, absolutely. Go ahead, and don't let anyone discourage you! Back in the day, the first solo-flight project a new java programmer would write would be something to suck porn from thehun.com . I applaud the maturity of our young programmers these days.
Yes, it is possible. But it seldom makes sense, because you would have to implement all needed functionality from scratch by yourself. This takes a lot of time and effort and often yields worse code than you would get by using a framework. Learning a framework also takes some effort, but once you learn a framework, you can easily use the knowledge on your next project.
I would go with Spring MVC 2.5+ using annotations. Very simple.