As mentioned, you can check on your own for upgrade inventory ("A" for the domestic legs to First Class, "C" for the international legs to Business class), or you can just call and ask. If you actually book the ticket with the agent, yes you will pay the fee. The risk of the upgrade inventory disappearing between the time you originally check it and then book the ticket online and then call to have the itinerary upgraded is small, but there is that risk.

Likely the only reason to book the "Saver" is that it probably is in a fare class that earns more Elite Qualifying Points than the super-saver. But it would certainly be unethical (IMHO) to make your company pay an extra $700 for that benefit. The Flexible ticket obviously saves you on miles & co-pay, and also provides for penalty-free changes/refunds. There is a large premium charged for the flexibility, obviously, and most companies prohibit or severely limit the booking of this type of ticket. Again, it would be unethical to buy the flexible ticket for the sole reason of reducing your co-pay/mileage requirement. But if you really do need that flexibility (i.e., there is a chance the trip gets canceled) and your policy allows it, then you'd get to benefit from the lower co-pay/mileage.

Another question is whether your company will reimburse you for the upgrade co-pay cost -- I think it's the least they could do consider they're requiring you to fly 12+ hours each way in a small cramped seat on their behalf, and it's still significantly less expensive than paying for business class. But you'd have to ask.

Well, you can try pricing out the itinerary on alternate sites and comparing. I would also try both AA's "search by schedule" and "search by price & schedule".