Steady state current in circuit containing dependent source
whether dependent sources always acts as a resistor (absorb power)

Consider the below simple example. The equations are
\$\frac{d i_{L1}}{dt} = \frac{1}{L1} V_{CCVS1} = \frac{1}{L1} \cdot 1 \cdot i_{L1}\$
This leads to an exponentially increasing current which goes to infinity (if initial current is not zero); i.e. no steady state.



simulate this circuit &ndash; Schematic created using CircuitLab

whether dependent sources always acts as a resistor (absorb power) or it can also deliver power

Oscillators designs use negative resistance elements along with LC(R) circuits to deliver power to compensate for the power dissipation in the LC(R) circuit. The negative resistance is often modelled as a dependent source (eg. BJT models). A example from Wikipedia showing the dependent source in a BJT model is shown below.


or it can also deliver power so that steady state current can become some finite value also?

Flip the + &amp; - symbols of the \$6 \cdot i_1 (t)\$ dependent source in your circuit and try to find the solution. Like the example shown above, the source will now try to increase the current in the coil; i.e. supply power to increase energy stored in the coil.