The path of Vo2 (2 OPA) is obviously longer than Vo1 (1 OPA). Why there is no synchronization problem between them? Are these OPAs fast enough so we can ignore the asynchronous?
This IC is for audio signals so the opamps do not need to be that fast to be "fast enough". Look at the datasheet, page 8, figure 26 which shows the open loop gain over frequency. At 0 dB gain (Av = -1) the Bandwidth is more than 3 MHz. That means a delay of about (-3dB point at 3 MHz (T = 333 ns) so 90 degrees phase shift equals (333ns / 4= 83 ns) ) 83 ns.
The highest audio frequency is 20 kHz which has a period of 1/ 20 kHz = 50 us.
That 83 ns is only 83 ns / 50 us = 0.17 % time error, at 20 kHz. In my view, no one can hear that.
So yes, the opamp is fast enough for the signal delay not to be a problem. Since we're dealing with audio, it is not an issue.
Why there are different design of BTL. What is the difference between them?
The idea behind a BTL is that a load is driven by outputs that are in anti-phase. There are several ways to implement this. It depends on the application and the choice of the designer what circuit will be used. For example some BTL chips are designed such that they can be used as a BTL driver or two non-BTL amplifiers for example to drive two speakers (at reduced power) for stereo. Each design has its own advantages and disadvantages.