Hi, I am an international student studying in the US too. I will be pursuing an Applied Mathematics PhD in Fall 2018.
In my opinion:
1. You should treat the research with your professor seriously.
Even if it does not produce any paper, it is fine. Work hard on it and everything else will come naturally. For example, for me, I did a research project with my professor in my last semester of study. It ended up bringing me to two conferences for poster presentations (There are actually a lot of conference opportunities... At least more than I would have thought of!), and a strong recommendation letter from him. You could aim the same. A strong recommendation letter from a professor who you do research with would be very valuable!
2. Looks like you have a very good GPA! Try to maintain it, as I think a high GPA is very important too. Also, I think you have excellent academic credentials already (with the honors classes and the grad level classes), at least compared to mine...
3. Maybe invest in a whole summer studying for the general GRE and try to get a good grade on it. Also, you will need to invest in at least a few months to prepare for the Math GRE to get a good grade too (I didn't take the Math GRE myself, so just some suggestions). So plan your time out
accordingly, especially when you also need to focus on a lot of other things too.
4. Identify potential people for your recommendation letters! This should come naturally from your research, classes, and any math-related job as well. You can talk with them more so they can get to know you more. Ideally, you should have professors writing for your recommendation letters for a Mathematics PhD program.
5. As grad school is not only about research, but also teaching, you may consider taking on some mathematics teaching/tutoring/mentoring jobs to get the experience.
6. Personally, I think it is fine if you don't know what mathematics you want to do research on in the future yet. They are all so interesting, so it is hard to choose really. My professor gave me the advice that I should keep my mind open, and decide on what I want to study after I am exposed to more mathematics. But anyway, I think mentioning a few things/directions that you want to pursue in the future in your SOP would be very helpful. Some schools are strong in a certain areas, and if you are interested in those areas it would be a good research fit.
7. And if you study in the US for undergraduate (4 years), you don't need to take the TOEFL!
8. When you are applying, make sure to ask your professors for advice on the applications (eg. school selection, SOP, CV, etc)! They are much more experienced than us and could offer a lot of helpful information. Also, see if there are any resources in your university on grad school applications, like a workshop on how to write your SOP/CV,etc. You can also consult grad students in your university on how they applied for grad schools. Ask more people! Usually they are very willing to help, and their information is very valuable.
The above are just my opinions! Please based on your own situation to decide on what to do next.