DAC + op-amp buffer just keeps dying
I'm have a hard time interpreting your question. Could you clarify the following?


  What happens in my case is that when I connect the circuit, only the connected DAC channel stops working - it doesn't give any voltage out anymore.


From the DAC Datasheet:


The DAC outputs voltage, so you can identify your problem by building your circuit in steps.
1. Set up DAC. Is it outputing the voltage you expect?
2. Set up op amp follower. Operating as expected?
3. Set up stage 2,
4. Set up stage 3,
5. Attach transistor.  

This will help you localize you problem to a circuit component or sub-circuit.
Finally, if your singals are &lt;10MHz, you might want to consider building this on a breadboard before soldering it.