Not a current student, but...
1) Your best chance of matching will always be at your home institution or your home state. That being said, the disastrous situation caused by Hurricane Katrina will forever associate New Orleans with "emergency" in my mind. I don't think you will be at a disadvantage if you pursue EM research, do away rotations, and network at EM research conferences.
Furthermore, there's a
Tulane Emergency Medicine Interest Group | Tulane University School of Medicine that will help with your networking and learning about EM opportunities at Tulane. And finally, if you take a look at the
https://medicine.tulane.edu/sites/medicine.tulane.edu/files/pictures/Match 2018 Web Results 2018 03-15.pdf you will see that 13 students matched EM. It seems like LSU SOM has an EM residency program, so if you want to stay in New Orleans for your EM residency, you could do an away rotation there and just network there in general. So it's definitely possible!
2) I personally find it a non-issue. The cleanliness of an anatomy lab will not affect how well you learn the material. Furthermore, you can always study in a coffee shop or at your apartment if you don't like how the facilities look.
3)
DeBakey Scholars Program | medicine or you can reach out to professors and ask about research opportunities.
4) Cannot answer this, but I remember being told that you can access the Undergradaute Gym with your medical student ID card. Keep in mind that the Undergraduate campus is in Uptown while the medical school is in the downtown area, very close to the French Quarter.
Hope this helps!