Wien's fifth power Law and Stephan Boltzmann's fourth power laws of emissive power Wien's fifth po

Deeper underlying explanation for color-shift in Wien's Law?
Suppose we have a blackbody object, maybe a star or a metal (although I understand neither of these are actually blackbody objects, to some extent my understanding is they can approximate one). According to Wien's Law, as temperature increases, peak wavelength will decrease, so the color we observe will "blueshift." Despite having reached Wien's Law in my research, my understanding is that this is in fact not an answer to why the blueshift occurs, but just a statement that the blueshift does occur.
So, is there some deeper reason why blackbodies peak wavelength emitted decreases with temperature?

What is the most intense wavelength and frequency in the spectrum of a black body?
Planck's Law is commonly stated in two different ways:
$u_{\lambda }(\lambda ,T)=\frac{2h{c}^2}{{\lambda }^5}\frac{1}{e^{\frac{hc}{\lambda kT}}-1}$
$u_{\nu }(\nu ,T)=\frac{2h{\nu }^3}{c^2}\frac{1}{e^{\frac{h\nu }{kT}}-1}$
We can find the maximum of those functions by differentiating those equations with respect to $\lambda$ and to $\nu$, respectively. We get two ways to write Wien's Displacement Law:
${\lambda }_{\text{peak}}T=2.898\cdot {10}^{-3}m\cdot K$
$\frac{{\nu }_{\text{peak}}}{T}=5.879\cdot {10}^{10}Hz\cdot K^{-1}$
We see that ${\lambda }_{\text{peak}}\ne \frac{c}{{\nu }_{\text{peak}}}$. So what frequency or wavelength is actually detected by an optical instrument most intensely when analyzing a black body? If they are ${\lambda }_{\text{peak}}$ and ${\nu }_{\text{peak}}$, how is ${\lambda }_{\text{peak}}\ne \frac{c}{{\nu }_{\text{peak}}}$?

Uses of Wien's law of displacement
Wiens's displacement law says
${\lambda }_{\text{max}}T=\text{a constant}$
So if I have the ${\lambda }_{\text{max}}$, I can find the temperature of a star. But if I have the temperature, is there any point in calculating ${\lambda }_{\text{max}}$? What information does that give us of the star, besides temperature?

Derivation of Wien's displacement law
I know this might be a silly question, but is it necessary to know Planck's Law in order to show that ${\lambda }_{max}\propto \frac{1}{T}$? If you set
$u(\lambda ,T)=\frac{f(\lambda T)}{{\lambda }^5}$
then
$\frac{\mathrm{\partial }u}{\mathrm{\partial }\lambda }=\frac{1}{{\lambda }^5}\frac{\mathrm{\partial }f}{\mathrm{\partial }\lambda }-\frac{5}{{\lambda }^4}f=0$
$\frac{\frac{\mathrm{\partial }f}{\mathrm{\partial }\lambda }}{f}=5\lambda$
But I am stuck here because if I integrate the L.H.S.
$\int \frac{\frac{\mathrm{\partial }f}{\mathrm{\partial }\lambda }}{f}d\lambda =\log (f(\lambda T))=\frac{5}{2}{\lambda }^2$

How can Wien's Displacement Law be 'changed' to a version for frequency?
Wien's Displacement Law stated that for a blackbody emitting radiation,
${\lambda }_{max}={\displaystyle \frac{1}{T}}$
where $T$ is the temperature of the body and ${\lambda }_{max}$ is the maximum wavelength of radiation emitted.
Due to the relationship between wavelength, frequency and the speed of light, a value of maximum wavelength would give a value of minimum frequency, and vice versa.
I then saw on the Wikipedia page for Wien's Displacement Law that
$f_{max}={\displaystyle \frac{\alpha k_{B}T}{h}},$
where $\alpha =\mathrm{2.82...}$, $k_B$ is Boltzmann's Constant, $T$ is the temperature of the body and $h$ is Planck's Constant.
How can this relationship for maximum frequency be shown?

Area under Wien's displacement graph
Why does the area under Wien's displacement graph give Stefan-Boltzmann law for a black body?
I couldn't find any proof of this. (I could just find this expression). I am not aware of the function of Wien's displacement graph as well (I just know that it is between Intensity and wavelength emitted by a black body).
Is there a mathematical way to prove this?

Showing Wien's Displacement Law from Wien's Law
Does anyone know how I would show that $\lambda \ast T$ is constant, using only Wien's Law? That $\rho (\lambda ,T)=1/{\lambda }^5\ast f(\lambda T)$ I differentiated, but all I could get was $\lambda T=5f(\lambda T)/f^{\prime }(\lambda T)$, which I don't think means it's necessarily a constant.