Is there any short way of seeing questions that I could possibly answer? (e.g., based on the tags of my own questions or something)
The SO homepage uses a weighted algorithm to try to focus on unanswered questions that you are likely to be able to answer. It places priority on Favorite Tags (can be added via the upper right corner of the homepage) and tags which you have provided good answers for in the past.
See this blog entry for an explanation of the actual algorithm.