Step 1: Recall the lac operon parts.
The lac operon of E. coli has a regulatory gene $i$, a promoter $p$, an operator $o$, and structural genes $z$, $y$ and $a$.
Step 2: Check option 1.
The true inducer is lactose (its isomer allolactose), not galactose. Galactose is just a product of lactose breakdown, so this statement is wrong.
Step 3: Check option 2.
The $i$ gene is transcribed and translated at a steady baseline level all the time, regardless of lactose. This continuous baseline is called constitutive expression, so this statement is correct.
Step 4: Check option 3.
Lactose actually inactivates the repressor by changing its shape, so the repressor cannot bind the operator. It does not activate the repressor, so this is wrong.
Step 5: Check option 4.
The $i$ gene has its own separate promoter, while only $z$, $y$ and $a$ share one common promoter. So saying all four share one promoter is wrong.
Step 6: Conclude.
Only option 2, that the $i$ gene is constitutively expressed, is correct.
\[ \boxed{\text{Gene i is constitutively expressed}} \]