Why is the output of the LPF half that of the real/imaginary part of the complex envelope?
The idea of this exercise is to show what happens if you have a frequency offset at the receiver, i.e. if the actual carrier frequency is different from what the receiver thinks it should be.

You can write the passband signal as

$$u_p(t)=\text{Re}\left(I_{[-1,1]}e^{j400\pi t}\right)\tag{1}$$

Since you're demodulating with a frequency of $\omega_0=401\pi$, it is advantageous to rewrite the passband signal with that reference frequency:

$$u_p(t)=\text{Re}\left(I_{[-1,1]}e^{-j\pi t}e^{j401\pi t}\right)\tag{2}$$

From $(2)$ you can read off the new complex envelop with respect to the actual demodulation frequency $\omega_0=401\pi$:

$$u(t)=u_c(t)+ju_s(t)=I_{[-1,1]}e^{-j\pi t}=I_{[-1,1]}\cos(\pi t)-jI_{[-1,1]}\sin(\pi t)$$

Demodulation with $\cos(401\pi t)$ will give you the I component $u_c(t)$. The factor $1/2$ simply occurs because by multiplying with a cosine you get $u_c(t)\cos^2(\omega_0t)=u_c(t)(\frac12+\frac12\cos(2\omega_0t))$, and the LPF removes the component at $2\omega_0$, which leaves you with $\frac12 u_c(t)$.