1) Does the 6143 fulfill the timing requirements ?
Yes, the 6143 should fulfill these requirements. It has a 250 KS/s sampling rate per channel which should give you 5x oversampling at the 50 kHz maximum bandwidth you indicated. The minimum delay from the recognition of the trigger to the first sample on this board is 100 ns which falls within the 900 ns window.
2) Is it posible and easy to set the board to take one sample after the trigger ?
It's not clear to me whether you want to take a single sample after a trigger and stop or use the trigger as an external sample clock to acquire several samples at the rate of the trigger. In either case, it's possible and relatively easy to do so. The only quirk is that the minimum number of samples to acquire for a hardware timed acquisition is two samples. This means if you really only want one point after the trigger, you'll have to setup the hardware to acquire two samples and then throw the second one away in software.
3) Is there a better choice than the 6143 to solve the problem ?
This seems like a pretty solid choice for the application you described. However, you didn't mention what the amplitude of your input signals are. The 6143 has a fixed +/- 5 volt range. If this isn't sufficient, you'll have to provide signal conditioning or choose a board with a wider input range. Also, I'm assuming your trigger signal is a TTL signal. If it's an analog signal, you'll want to choose a board with an analog trigger circuit (the 6143 doesn't have an analog trigger). The PCI-6111 is a 2 channel board with a +/- 42 volt input range, an analog trigger, a 5 MS/s sampling rate, and will also meet all of your requirements. The only down side of the 6111 is that it is more expensive and is a 12 bit converter instead of 16 bit. The PCI-6120 has even more functionality than the 6111 and has a 16 bit converter at 800 KS/s. PXI has some more simultaneous offerings, but I'm assuming you want to stick with PCI. You can also check out some of our modular instrument offerings such as the PCI-5102. I'm not as familiar with the modular instruments, so you'll have to check out the specs for yourself. Whether there is a better choice or not is pretty subjective. That will depend on how much you're willing to spend for the features and specs you really care about. From the information you provided, the 6143 will be the lowest cost solution that meets the requirements you specified.
4) Can we solve this problem programming in VB or VC++ and using the software that comes with the board or do we need to buy extra libraries ?
The DAQmx driver provides interfaces to both of these programming environments at no extra cost.