Welcome to FlyerTalk,
joedev.
If yu will be flying the oneworld alliance in your future, the Challenge is a fast track to mid-level (AAdvantage Platinum / oneworld Sapphire) status; the members' guide can be downloaded in PDF format
here.
One important thing about the Challenge - and I heartily recommend you read more about it in the Challenge Help Desk thread
here and in our FlyerGuide AA Wiki
here - is it is based on POINTS, not miles flown.
How many EQ Points you earn will depend on the fare - $1,600 doesn't tell us much; some highly discounted fares earn a half-point per base mile / mile flown, so your itinerary - LAX-LHR-GVA-LHR-LAX renders 11,852 miles flown / base miles. For Coach:
Highly discounted fares (G, Q, N, O & S) = 5,926 points, not sufficient for a Challenge
Discounted fares (H, K, M, L, W & V) = 11,852 points, sufficient
Full (Y and B) coach fares = 17,778 points (also for Business / First)
See the table and explanation
here.
The Help Desk thread explains, among other things, how you can put a booking on hold (you can see the fare class when you do that and see the detailed fare rules), call Web Services, upfare to a fare than offers 1.0 point per mile if you need to do so, and then complete the purchase on aa.com.
I believe AA is offering the new 777-300ER on LAX-LHR currently - it would offer Main Cabin Extra, so you might look at what you can do to purchase those seats - extra legroom, and one less seat across than in plain coach. (Yep, there's a thread for that...)
And sometimes one can find discounted "I" Business fares for not much more than some coach fares.