Math major here. I've talked at length with my department's graduate chair, and have helped him organize workshops on the topic of math graduate school. We are not a tier-1 school so ignore this advice if you don't think it applies to your unique situation.
-Whether the application is submitted on time. For my school, it's not super important because we don't get a ton of applicants. For the more competitive programs in my university, late apps are not even considered. If you're really concerend, you can simply call the departments you applied to and ask them if they accept late apps.
-Whether GRE general test have been taken. For my department, the GRE is not required. However, not submitting one counts against you, even if your scores are mediocre. The way my chair explained it was that each piece of your application package gives him an idea of who you are. When a piece is missing, it leaves a vacuum that must be filled by other things like your GPA and LoR. Therefore, your GPA and LoR must work harder than if you submitted a GRE score.
-Not stellar GRE maths subject test. This one is important. Grades give some indication of your math skills, but a high GPA only says that you are good at getting A's, not that you have any math knowledge.
-No REU. Not a big deal. But having a REU on your application makes you stand out.
-Only have 3 letters of rec. This is standard. Submitting more letters than required can count against you. It depends on how busy the application screeners will be. They are most likely swamped by applications, and would not appreciate having to process extra LoR.
-Whether recommenders are strong researcher. What do you mean by "strong" researcher? LoR are most meaningful from people who are respected in the field, who have worked with you closely (rather than someone who was your prof for one semester), who are full professors (rather than lecturers or adjunct faculty).
-Undergrad is in liberal art instead of univ. Getting your undergrad from a more prestigious school helps, but it won't hurt your application if you did not.
-Lacking graduate level maths course. Not a requirement but this would make you less competitive. All the undergrads I know who are considering math graduate school all have several graduate level courses under their belt. Keep in mind that we're not even a prestigious math program. I imagine that the people competing for spots at tier-1 schools would be at the top of their game.
-Lacking basic maths course. This one is bad. At the very least, you should have a strong foundation in analysis and algebra. My program will overlook this requirement for applicants to the master's program if the rest of the application is strong, but the student has to take these courses since they are pre-reqs for graduate courses.
-Low cumulative GPA. GPA isn't as important as the other part of the application package, but my university won't consider graduate student applications with cumulative GPAs lower than 3.0 unless the department can justify why they want to accept the student. Your application would have to be very compelling for the department to jump through those hoops for you. Other schools will have different policies, I'm sure.
-Failed/low grade in many non-math course. Any failed courses is cause for concern, but an F in a non-math courses isn't a dealbreaker if the rest of your app is strong, especially if the bad grade was from freshman year. Again though, this is for my school and the level of our applicants are not in same league as MIT. Most of my fellow math undergrads have 3.8-4.0 gpas, and we are required to take a LOT of classes outside of math to fulfill our general education requirements. "Many" bad grades, even in non-math courses, looks really bad.
-Some maths course with not perfect grade. B's are okay. C's less so. B/C's in lower level classes like calculus aren't as bad as B/C's in upper level classes.
Anyways, I hope that helped, and good luck on your app!