What is the grey film/black sludge from mercury? How to remove it?
First, I would advise against using mercury in a home lab due to the ease which scattered droplets fall into cracks and evaporate. 

That said, if distillation is out, then try bulb-pipetting from the center of the mass of mercury to avoid some of the surface contamination. Avoid dropping the mercury when lifting the pipette, since $\ce{Hg}$ is so dense it tends to run out as soon as lifted. Ideally, work in a hood with a large tray with raised sides underneath the apparatus.

If purity is important, electrolysis will yield a higher-grade product... at the risk of working with even more toxic compounds!

Better, get some gallium if you want to play with a liquid metal, and don't need $\ce{Hg}$ to make Grignard reagent or other esoteric substance.