Do we need to check in 3 hours in advance if we're flying business class to the U.S. And has anyone had experience using the AA self-check-in system?

AA's website recommends 2 hours. The website also has information on self service machines and online check in for international flights.

I would think 2 hours should be plenty of time. My bigger concern would be not cutting it too close on the trip to the airport. You don't want to be sitting in traffic watching the clock tick down.

Remember that biz will mean lounge access, so extra time at the airport may pass pretty fast and make better-safe-than-sorry a good option.

I've always found 2 hours was more than adequate for AA business class check-in at LHR.

The lineups at the business class counters are seldom very long. Note that they require a bit of finding - back in the corner.

Be sure to allow enough time to get from the business class lounge to the gate. Believe the signs about how long the walk takes. We had to really hustle to make our BA flight last year, just too comfy in the lounge.

Does anyone think that the same will be true for BC on Air France?

you need to check the AirFrance website. Each airline has their own rules/requirements.

I would allow an extra half hour just to be on the safe side.

If you want time to swan around the shops and you want to relax in the lounge, then check in 3 hours ahead. If you to spend minimal time in either the shops or the lounge, two hours is fine.
If you're flying over a holiday or other peak period, go for the 3 hours.