Somewhere at least as well ranked. Most schools make their new hires from peer institutions and better ranked ones.
Insanely stupid. Many schools have a policy against hiring their own PhDs. If your dream job is to work at South Carolina, read the CVs of their faculty (especially the junior ones) and look at where they went to school. How many of them have degrees from South Carolina? The most common schools are where you should be looking. My own quick scan showed that currently no members of the USC faculty received their doctorate there. The modal degree was from UNC. I also saw multiple graduates of Cornell, Indiana and Iowa.
Depends entirely on the research agenda you ultimately create for yourself.
Its going to hurt you applying to some schools. If you read the posts by BFB on Ohio State's admission's process, you'll note that the have to put the people they want to admit into a school wide competition for funding with a cut-point for undergrad GPA. You will be in trouble at schools that do that. Other schools give their department's an allotment of fellowships to give out to the applicants they want. At those schools, they will be able to weigh the later performance and make up their own mind.
A better way to think about GRE scores is as a threshold. As far as I know, departments aren't admitting people because of stellar GREs but they will reject someone for poor ones. They all have some threshold. Once you get over that, how much you are over by is not a very big deal. Again, see posts by BFB for how he interprets GRE scores. Applying for a PhD program where departments are making in the ballpark of 10-30 offers is very different than applying to law schools where places make hundreds of offers and you can explain the majority of the variance in admissions using just LSAT and GPA.
Again, variable. Depends a lot on what you did with the time. Some places will not like it. If you used it productively and got some insights into what you want to work on, then can be a plus.
Hell if I know, but there are at least a few people posting on this forum who have been on admissions committee or three.