In animals the sensory capabilities evolved over hundreds of millions of years. It is therefore not surprising that different sensory aspects use similar transduction mechanisms.
a) In terms of transduction (i. e., the way energy or chemicals are sensed and transduced into an electrical signal), which distinct human sensory systems use the same molecular mechanisms?

Olfactory system

Explanation:

In an olfactory system, odor molecules are perceived by the olfactory receptors. These odor molecules are nothing but the chemicals which are then transduced as chemical signals into the electrical signals and sent to the brain. The brain perceives these electrical signals as smell.  

The odorant particles bind to the specific receptors located at cilia which then signal through the G protein Gαolf which then activate adenylate cyclase. Adenylate cyclase leads to cAMP production which then open a cyclic nucleotide-gated ion channel through which both sodium and calcium ion entern into the cell. Calcium ion then activates chloride channels and thereby causing transduction of electrical signal