The gunk at the bottom of the bottle is from adding sugar to the bottle for carbonation, which isn't bad. Like the others said, you pour all but 1/2 inch of beer at bottom and your ready to drink. You could always go to a keg and force carbonate. It shouldn't be more than a one time cost of $70-80 for a keg and $50-90 for the Co2 tank. It pays for itself after about 150 bottles, or 6 cases of bottled homebrew.
Whirlpooling is a must, and it's easy.
The clarity of beer has a lot to do with floating proteins and suspended yeast. Proteins don't affect the taste, but yeast does. A lot of people remove yeast by "crashing the beer" (lowering the temp) to the point of freezing for a few days, rack off the beer and bottle/keg.
If there is any remaining haze (typically at cold temps), it's from floating proteins. Using the Irish Moss certainly helps, but a strong boil along with adjusting the pH balance in the mash, whirlpooling and a clarifying agent will clear up the beer quite a bit. My favorite clarifying agent is Polyclar along with a size 5 water filter. My beer is typically brilliantly clear. Filtering Polyclar is recommended due to the use of Silicone, which is not recommended for ingestion by the FDA. However, most homebrewers rack off the top of the Polyclar without issue.
Cheers,