[Advice on gettting into pure math grad schools]

Northwestern does not require Math GRE. It also has a great graduate program with very strong faculty in dynamical systems, geometric analysis and probability. Applied Math is in a different department. They got several NSF post-docs out of their program recently. 

[Recommendations for universities with strong probability faculty.]

I like your choices but you are missing a few excellent places that are worth a shot. If you are interested in probability/stats, I would remove Purdue/TAMU/Indiana/UCI from your list. I would add Courant/Chicago/Northwestern. In probability, they beat most of the schools in your list in terms of pure research quality as they have incredible faculty in the field (Newman/Bourgade/Lawler/Smart/Auffinger/Zelditch/...). Their grad students have a record of finding top post-doc positions. Furthermore, location-wise it is hard to beat NYC and Chicago/Evanston if you like to live in a big city. 

In OR, you may want to consider MIT and Princeton.

[Need advice on choosing math/probability PhD program.]

I like your choices but you are missing a few excellent places. If you are interested in probability/stats, I would remove Purdue/TAMU/Indiana/UCI from your list. I would add Courant/Chicago/Northwestern. In probability, they beat most of the schools in your list in terms of pure research quality, they have incredible faculty in the field (Newman/Bourgade/Lawler/Smart/Auffinger/Zelditch/...) and their grad students have a record of finding top post-doc positions. Furthermore, location-wise it is hard to beat NYC and Chicago/Evanston if you like to live in a big city. 

In OR, you may want to consider MIT and Princeton.