Is the statement "The deactivation sensor sends a signal to disarm the bomb" frame-independent in the train-tunnel paradox?

Yes. It is frame-independent. From the tunnel frame it is obvious that the signal is sent, as this happens before the bomb explodes in that frame. From the train's frame, the bomb explodes before the signal is sent, but that doesn't mean the signal is not sent. This is because the bomb is at the front of the train, therefore not all the train explodes at the same time. When the bomb explodes, the front of the train explodes, but the end of the train (and therefore the sensor) is yet perfectly still, because whatever destruction the sensor may suffer due to the shock wave produced by the explosion takes at least a time $$L/c$$ to reach the end of the train. On the other hand, the time that it takes the back of the train to enter the tunnel since the activation of the bomb is $$\displaystyle\frac{l_{\text{tunnel}}-L}{v}=\frac{\displaystyle\frac{L}{\gamma}-L}{v}=L\ \frac{1-\gamma}{v\gamma}$$ and you can check that this time is less than $$L/c$$, and this means that the signal is sent before the sensor is destroyed by the wave caused by the bomb.

Is the statement "The deactivation sensor sends a signal to disarm the bomb" frame-independent in the train-funnel paradox? If yes, what is the correct answer to the above statement?

Yes, it is a frame independent statement. You can think of it as a measurement. The sensor measures something and reports the value of that measurement. The outcome of any measurement is frame independent, meaning for example that I may observe your watch to be running slow, but both you and I will agree on what your watch actually says. You will believe that it is running correctly and I will believe it is running slow, but whatever it reads we both agree.

As described the sensor sends the signal. The signal is sent whenever the back of the train passes near the end of the tunnel. The back of the train passes near the end of the tunnel in all frames so the signal is sent in all frames. The bomb explodes before the signal is sent in the train frame, but the explosion travels (much) slower than c and therefore the deactivation sensor in the rear is still intact as the rear reaches the end of the tunnel.