Might be best to first focus on the signup bonuses hinted in the other responses here. Even with your spending level, you could earn far more from signup bonuses for the next year or two than from actual spending.
Between the two of you, it should be easy to accumulate 468k miles in AA within 6 months, using $18000 of spend. With AA, if you redeem in coach, it could be as low as:
Mexico - 25k round trip
Hawaii - 35k round trip
Europe - 40k round trip
In other words, if you redeem in off-peak times to Europe, just the CC signup bonuses could be enough for 5 trips for two people. There is a thread in MilesBuzz! about the AA CC bonuses.
As far as a day-to-day spending card, you would probably do better with one of the airlines that offers category bonuses (like 2 miles/$ in groceries) or perhaps a 2% cash back card. But that difference is minuscule compared to signup bonuses available.
To answer your general questions:
- It's usually easier to redeem on a partner airline NOT based in the US. For example, if you have Delta miles, it's usually much easier to find availability with Air France. Just have to make sure you pay attention to the taxes/fees, since some routings will impose a hefty fuel surcharge.
- Partners usually show the same availability that they show directly to their own members, but usually only "saver" awards, not highest-mileage "standard" awards. One notable exception is United, which sometimes artificially blocks availability it shows its members from its partners.
- If you want to get serious about finding seats, you should look into the
KVS Availability Tool. It has a bit of a learning curve but there's no better/faster way to identify availability on a variety of airlines.