Tickets are always priced in the currency of the origin airport, which is why it is in Yen. In terms of charging to Amex, it will show up on your statement as the amount in Yen under the Foreign Spend column, and then converted to USD in the Amount column. I don't believe any of the Delta AmEx cards have a foreign transaction fee so you shouldn't need to worry about that.

The conversion should be done at whatever the rate was at the time of booking the ticket, although it is possible it is not done until it goes from pending to a confirmed charge. Someone else may be able to chime in on the exact timing of when the rate is locked in.

There is no way to get the ticket to price out in USD and I certainly wouldn't play the guessing game on when/if the conversion rate will be most in your favor. Unless you're looking really far in advance, you're more likely to pay more due to fare changes than you would trying to time up the best forex rate.