Mili,
I think you'll have to experiment, but it might be possible to do both operations with the equipment you have. This would be relatively easy with an NI 660x counter/timer board.
You can use the Route Signal VI to send your counter output to a RTSI line and then select the RTSI line as your counter 1 gate. (this is the generic solution for any number of counters)
For getting an external update clock, check out the continuous external clock example programs that are included with LabVIEW or NI-DAQ.
You can easily start your output with a digital trigger, but the STC counters do not allow for digital triggering of counter operations. You may be able to come up with some programming scheme that gets counter 1 started, but gated low so t
hat it is not counting until the start trigger comes in for the acquisition (such as a start trigger that is forced low until the start trigger for the analog input comes in). This would be a design you would have to work out or see if anyone else has found a way to do it.
You can also search for many example programs/discussions at http://www.ni.com/support by searching in Option 3 for keywords like external, analog, output, and trigger.
Hope this helps.
Regards,
John Nieri
Applications Engineer
National Instruments