A: I recommend taking some of the core courses for Applied Mathematics Majors before getting involved in research. This way you are more likely to have the skills necessary to do an interesting project. Go and talk to professors about what courses might be required before working on a project with them. this may help you choose what is most appropriate.
A: No. If you continue to attend colloquia, over time you will become more familiar with the ideas and terminology. After some time you will begin to understand more of the lectures.
A: Go and talk to the professors you know about their research. Attend 290 seminars to better understand what the faculty member works on and what their students are working on. Talk to students of the advisor but do not use this as a substitute for talking to the advisor directly. The students typically know their own problem but not the full scope of research projects that the advisor might be interested to supervise.
A: First of all, you should go and talk directly to Prof. X rather than relying on information from friends. It's possible that Prof. X might be taking students. If not Prof. X can explain why and give you some advice about other faculty who might be supervising students on related problems.
A: Starting your first research job after PhD can be a tough transition point for many people, especially if you do not have a preassigned problem to work on. First off, you are likely to have been assigned a mentor by your department. You should go and talk to them and ask them if they have a suggestion for a problem to work on. You should also talk to other faculty about what they are working on and try to regularly attend seminars. Many research projects often get started from informal conversations with people. Also make sure you are budgeting your time well so that you have enough time to do research. If you spend all your time on teaching, in what is expected to be a half-teaching, half-research position, then you need to adjust how you prepare for your classes. Set goals for yourself in terms of productivity, on a daily, weekly, and monthly basis and try to meet those goals.
A: One important thing to remember is that the job market is an international one, for the most part. So your competition is not the people down the hallway, necessarily, but it could be someone on the other side of the country. At the same time, the pool of possible positions is large, so if you apply very broadly you are likely to get something good, especially if you are doing good work and coming from a good place. Also many schools are looking to hire someone who fills a void in their current faculty research interests, and this will vary quite alot from one department to another.