Hi LostLamb,
These are all great questions, and I'm interested in hearing the perspectives of others.
In general, the answers to your questions are all "it depends on the program and advisor". Programs range very widely in terms of the mathematical rigor, some programs leaning strongly towards the "psychology" side, with classes that need to be accessible to the full psychology department and encouraging quant psych students to become involved in applied research with others in the applied area of the department, and some programs lean strongly towards the "quantitative" side.
1. The level of mathematical rigor:
In terms of admissions: I've seen students accepted into programs from very different backgrounds and math preparations (when I was accepted into my program, I had a masters in math/stats... and the other girl that got accepted into my program had a stats minor in undergrad, I think she had taken Calc 1, but her experience was mostly in applied research). If you've worked with derivations and epsilon-delta proofs, your math background seems fine.
In terms of the courses taken during the program: it differs a lot by programs. A lot of quant psych programs are housed in psychology departments and need to be accessible to the other psychology branches (clinical, social, etc.), whose math background may be as little as a class or two in college algebra. So the mathematical rigor in these classes is minimal (for someone who's taken so many math classes as you say), unless you are taking additional classes from the math/stats department.
In terms of research: depends on the advisor. The people that I've worked with do simulation work mostly. Psychometrika is a more technical journal, so if you're looking to working with someone technical, I'd find professors that publish there. SEM and MBR are generally on the less technical side.
PIs and programs, like you've pointed out, vary greatly in terms of where they stand on the spectrum between "mostly applied" to "extremely technical" work. Reading through some of their publications will probably tell you where they stand. From what you mention, it seems like you're doing that already, and that with you're background you're probably understanding a lot of the papers.
2. My answer to this question mirrors my answer in (1). Most accepted applicants from the last couple of years that I know have less background than what you're mentioning, but it ranges a lot.
4. Yes, previous experience with substantive research does help a lot. Since great part of quant psych is about bridging the gap between psych (applied) and stats, we value people in the middle who are quantitatively oriented but can communicate with psychologists and have a passion for psychology. If there is no interest in psychology, then a pure stats program would be more appropriate. So yes, especially since your background seems to be so strong on the mathematical side, I think it is important to highlight as well that you are interested and have worked with psych applied research.
6. It's program specific. Some programs encourage students to get a masters in stats along the way, and other programs do not offer the option to get a masters in stats. If this is something you're interested in, you can find out through the program's website, looking from alumni CVs, or directly contacting people that are in the program already. In my experience, most students are happy to answer questions (we are a small community and get very happy when we find out people are interested in our field).