  The recently obtained hairy Kerr black holes, due to additional sources or
surrounding fluid, like dark matter, with conserved energy-momentum tensor,
have a deviation $\alpha$ and primary hair $l_0$, apart from rotation parameter
$a$ and mass $M$. In the wake of the \textit{Event Horizon Telescope}
(\textit{EHT}) observations of the supermassive black hole M87*, a recent surge
in interest in black hole shadows suggests comparing the black holes in general
relativity (GR) and modified theories of gravity (MoG) to assess these models'
differences. Motivated by this, we take on an extensive study of the rotating
hairy Kerr black holes, which encompasses, in particular cases, the Kerr black
hole ($\alpha=0$). We investigate ergosphere and shadows of the black holes to
infer that their size and shape are affected due to the $l_0$ and are found to
harbour a richer chaotic structure. In particular, the hairy Kerr black holes
possess smaller size but more distorted shadows when compared with Kerr black
holes. We also estimate the parameters $l_0$ and $a$ associated with hairy Kerr
black holes using the shadow observables. The inferred circularity deviation
$\Delta C \leq 0.1$ for the M87* black hole is satisfied, whereas shadow
angular diameter $\theta_{d}=42 \pm 3 \mu as$, within $1 \sigma$ region, for a
given choice of $\alpha$, places bounds on the parameters $a$ and $l_0$.
Interestingly, the shadow axial ratio obeying $1< D_x \lesssim 4/3$ is in
agreement with the \textit{EHT} results and thus eventuates in the hairy Kerr
black holes being suitable candidates for astrophysical black holes.
