Thanks for the great questions @ralphc-nyc!
1) Can you give an example of a problem that you solved and your process?
On a recent project I was working with a very specific set of requirements from the client about how data needed to be represented in an internal application for employees. I had presented a few design options for the client that were different from the current approach, which was a pretty basic table display. The client responded with more requirements that constrained the design options even more. After the meeting I got together with my development team and we worked out some different options together. I was able to iterate my design right away so we could see what various options would work best. As a team, we came up with a solution that met the client's requirements but was still easy for the user to digest a lot of data at once.
2) How do I become a better problem solver?
Learn how to recognize problems and understand what they are. Sometimes it's good to take a step back to process and think about the best way of tackling the problem. Observe how others do it and learn from them. Then practice your own approach to solving problems. Try out different methods and approaches and see what works best for you. Everyone has their own way of doing it, and your approach can change based on the problem you're trying to solve. It can take time, but the more you see others doing it and practice it for yourself, the more experienced you become at it:)
3) How do you determine what research methods to use and the amount of methods for a project? Is research the first thing for a project after the requirements or does it continue even throughout the entire process?
I used to think there was one "right way" to approach research, but I've learned that it really depends on a number of factors. What things are you trying to learn? Are there specific questions you need answered? Sometimes project budgets or requirements can also determine what types of research you do. Having a good understanding of what methods you can use will help you determine what might be the best approach for your project. For example, if you want to understand how a user performs a specific task, you may want to spend time doing a contextual inquiry, an in-person observation of a user in his or her environment. Again it really depends on the project as far as when you conduct research. Sometimes it's done early on and helps to define requirements, sometimes it's ongoing. It's all about what your specific project needs are and what you've determined you need to understand.