Food4Thought

Of course you do. The highlights of Axler's book will be taught in most econometrics sequences. What I meant is that you should take the advanced linear algebra course if your anticipated research area requires knowledge of theoretical econometrics, for example financial econometrics or structural IO. If you are interested in mostly reduced-form micro (development, health, labor, public, etc.), then a less advanced linear algebra course would probably be fine preparation.

If you already have several advanced math courses (w/ good grades) on your transcript, then B would be the most useful. Though, you should know the material from Axler very well if you intend on doing anything that requires advanced econometrics.

6 is becoming increasingly common.

"I want to do everything I can to get into the best possible program." That would be a good way to make clear that you're serious while avoid coming off as pretentious. In undergrad, I got the best grad school advice whenever I was honest about my ambitions. That said, be realistic. Don't have a B average GPA and say you want to go to Harvard.

Stokey, Lucas, and Prescott (SLP) and Ljungqvist and Sargent (LS) are the standard macro textbooks most programs use. The difficulty of the course would be more accurately evaluated by the competitiveness of your program and whether it has a reputation for producing macroeconomists. I had pretty zero background in macro when I took the macro core. If you have past experience with dynamic programming (or simply like doing it), you should do fine.

The first two of these suggestions are good. Most basic results in point-set topology and analysis are useful in pretty much every sub-field of microeconomic theory. Unless you know you are interested in networks, I think your time would be better spent in a rigorous probability or stochastic processes course (probability measures, definition of expectation, martingales, brownian motion, etc.). Dynamics and continuous-time methods are becoming fairly popular in applied theory. So, the chances that you use that material are much higher than something more niche, like graph theory or combinatorics. If you are interested in decision theory, you will need analysis, topology, and probability theory. But you might also look into taking some form of advanced linear algebra, one that serves as an introduction to functional analysis (Banach spaces, Hilbert spaces, Riesz representation theorem, etc.).

I want to second Chateau's comment on prioritizing experience over math right now, especially if you are interested in applied topics. Majoring in math with about an A- average shows that you are not likely to be one of those who just doesn't get it. This will be a plus for your application, but you're going to need to appeal to applied faculty if you are to sell yourself as having such interests. And that kind of appeal usually comes from experience with a thesis or a research project of some sort where you get your hands dirty and work with data. The summer internship should help with that. And regarding your concern about not finding people with similar profiles who didn't do something after graduation, I think it is becoming increasingly rare to come straight from undergrad if you don't have an already very competitive profile. In my cohort, there's only a few who didn't do masters or RA work.

Do not give much weight to this idea. Distance does not tell you much about collaboration unless the proximity is extremely close (Harvard-MIT-BU, Stanford-Berkeley, Duke-UNC, etc.).

If you are RAing at your dream school, then good job. Regardless of where you go, it will be an awesome experience that you will look back on fondly. For the purposes of maximizing your chances of getting admitted to that school (or any of the top tier schools, for that matter), then time is your biggest advantage. People often don't realize that if you work an RA for only one year, your supervisors are going to have only 4-5 months of work to write about. This is not ideal, because an RA is something that, in my experience, takes getting used to. It took me about 7 months to really get good at my job. So, if you are willing to stay for 2 years, then you will be doing much higher quality work, and your letter will be stronger because of it. I would advise you to take advantage of all the resources available to you. When you can, attend seminars that interest you. After you get to know your supervisors, you will be able to ask them if they can connect you to certain professors whose research interests are close to your own. When I was an RA, I had a few meetings like this, and I found them incredibly valuable. If you take a class, make sure to give it your all and definitely make sure to not overload yourself. I underestimated how tiring a full time work schedule can be. If you are not confident that you will stand out in terms of grades, you can always ask to sit in on classes. I stayed at my RA for only one year. Despite this, I still got top 20 offers and some attention from top 10 schools (i.e., waitlists, unfunded offers). If you have a good profile and are interested in your job and do good work, I expect you will have a reasonably stress-free application season and should get a few offers from the range of schools you're targeting.

I would pursue doing a thesis in the area that you are most interested in. I wrote my senior thesis on a theoretical topic. The learning curve was steep, but I found it to be an incredibly valuable experience. Also, I am skeptical of this view that one should write an empirical senior thesis on the grounds that it is more likely to be "better". Undergrads rarely learn anything beyond OLS and very basic estimation techniques. So, it is not like students writing empirical senior thesis topics are extremely well prepared either.

I think the best way to get over your fear would be to work through an easy textbook on the material you are scared of. To that end, I would suggest something like Analysis with an Introduction to Proof by Stephen Lay. It teaches basic real analysis (sequences, limits, continuity, etc.) at very gentle pace. It is a pretty short book, and you can find the solution manual to the fourth edition online.

You are taking more advanced classes that should give credence to your mathematical ability. I would focus instead on getting good grades in advanced courses and doing work with your professors to get the best possible letters of recommendation. If you are incredibly worried about your calc 2 grade, consider taking a probability course. A good grade in that class should ease any concerns about your ability to do integration.

I wouldn't say it is a complete waste of time. You could get lucky, especially if you think your letters of rec are going to be strong. That being said, I don't think top 10-20 is where the bulk of your applications should be. I have a friend from undergrad who had a similar profile when he applied (even had a C in the calc sequence like you). He received acceptances from top 40 places and was waitlisted at one top 25. He eventually got off the waitlist and received full funding. He got very lucky, but it does happen.

Topology will not give you a tremendous advantage in the first year micro coursework, if you are already comfortable with proofs involving continuity, sets being open, etc. But a course in topology would be beneficial in the sense that it is a hard class that will improve your ability to prove things and think abstractly. The material you would cover in an undergrad sequence in math stats would be pretty close to what you'd cover in the first semester of an econometrics sequence. So, if you just want prep for first year, I'd go with math stats.

I say put econ stats down and continue with the application as normal. Let Wisconsin decide if they want to reject you or not, don't do it for them.