Is this reason why we can let the current source be attached to a grounded?
In this question, someone shows me that these two circuit are
  equivalent.


I didn't say that it was equivalent - I made a simplification that allowed me to find the current through R4 which then permits finding the current through R2. However, it can be made equivalent with one extra feature (splitting a current source): -

In your original question, you were interested in finding current \$I_0\$. So, my method was perfectly valid. However, if you were interested in knowing the current flowing from voltage source V1 then you would have to replicate the current source like this (I3 in the red box): -



The above would be the "true" equivalent circuit.

I3 (4 mA) is now correctly drawn from V1 but, it did not affect \$I_0\$ and so, in the original problem, I didn't bother to include it as you were only solving for \$I_0\$.

Regards your second question, if you were interested in the power dissipated in R10 then you have to keep it in series with the current source and, if the current source moves (to make analysis easier) then the added resistor moves too.

However, if you do introduce a new current source (as per above), you don't need to introduce a new extra resistor (a la R10) in series with current source I3 (placed across V1). There can only be one R10 even if you duplicate the current source. A resistor in series with a current source does not affect that current.