Welcome OP from a fellow PHX flyer (albeit after drive down mountain) and fellow overseas short term missionary. That is why I got into this game-to save on missions trips.

Only longhaul option out of PHX is British Airways PHX-LHR. Great for paid fares but really bad for Avios redemptions due to insane fuel surcharges.

To really help with credit card advice:

-how much do you typically spend per month on credit cards?
-what cards do you currently have?
-rough idea of your credit score/report?

Blanket statement: general consensus is apply for multiple cards on same day every 95ish days.