What does the input sensor on your transmisson of a Pontiac G5 se 2007 do

The speedometer not working wouldn't be caused by an input sensor . The VSS - vehicle speed sensor is a direct input to the PCM , the PCM shares is info over the GMLAN data bus.
Speedometer
The instrument panel cluster (IPC) displays the vehicle speed based on GMLAN serial data message from the body control module (BCM). The BCM receives vehicle speed data from the powertrain control module (PCM)/engine control module (ECM) via the GMLAN serial data circuit. The speedometer will default to 0 km/h (0 mph) if:
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The PCM/ECM detects a malfunction in the vehicle speed sensor circuit.

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The BCM detects a loss of GMLAN communications with the PCM/ECM.

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The IPC detects a loss of GMLAN communications with the BCM.

The only way to diagnose these problems is with a factory scan tool , this will check inputs an outputs at the PCM an BCM to the instrument cluster . This avoids guessing an playing the parts game. Your best bet would be to take your vehicle to a qualified repair shop .
Same thing with trans not shifting out of first gear , could be a shift solenoid ! Does it shift manually , moving gear selector through the gears ?