I only have time for a partial answer, so I'll discuss the bit about rotating the comet.
We have to do some calculations with radiation pressure, the force on an object from light.$^1$ This pdf has some good starting calculations, although they assume that the source is spherical - the Sun. Our laser clearly is not. However, the beam will diverge over such a long distance, so I'll have to use a substitution (given by CountIblis).
The force due to radiation pressure is
$$F_{\text{rad}}=\frac{2I}{c}A$$
where $I$ is the intensity, $c$ is the ubiquitous speed of light, and $A$ is the area onto which the force is applied. However,
$$I=\frac{\text{power}}{4 \pi r^2}$$
so we get
$$F_{\text{rad}}=\frac{\text{power}\times A}{2 \pi r^2c}$$
The $4 \pi$ has to be replaced (by CountIblis' substitution) with $$\Omega=\pi \left(\frac{\alpha}{2}\right)^2$$
For an ideal laser, $$\alpha=2.44\frac{\lambda}{d}$$
where $\lambda$ is the wavelength and $d$ is the initial beam diameter. Let's assume that our laser has a power output of one megawatt - the COIL laser, from a Boeing YAL-1, as per Randall Munroe's idea. $1,000,000 \text{ watts}$ is a lot from a laser. Let's also assume that no power is lost.
For COIL, $\lambda=1.315 \times 10^{-6}$ and $d=0.1016 \text{ meters}$. This gives an $\alpha$ of $3.158070866 \times 10^{-5}$, and thus an $\Omega$ of $7.83309915 \times 10^{-10}$.
Philae is currently on comet 67P/Churyumov-Gerasimenko, which has, on its longest side, a surface area (combined lobes) of about $19,370,000 \text{ meters}$. However, it's about $310,000,000 \text{ miles}$ from Earth, which is $4.9879 \times 10^{11} \text{ meters}$. Knowing that, to rotate the comet, we have to apply force on only half the area of one side, we find that this laser would apply a force of
$$F_{\text{rad}}=\frac{1,000,000 \text{ watts}\times 9,685,000\text{ meters}^2}{7.83309915 \times 10^{-10} \times 2.487914641 \times 10^{23} \text{ meters} \times 299,792,458 \text{ meters/second}}$$
$$=1.65771483 \times 10^{-10} \text{ Newtons}$$
Yep, we're not rotating a comet any time soon. I'll try to add in the bit about recharging Philae by laser sometime later. I do think that will be a lot more plausible.
$^1$ Unfortunately, I can't find any calculations on laser propulsion.