Help! I've pitched the yeast and my wort is not doing anything.

The first phase of fermentation is yeast reproduction. This is the "lag" period when nothing appears to be happening. Once sufficient yeast has been produced, they begin to eat the sugars and fermentation is evident by slight white dots and foam that soon builds into a large, foamy head, or "krausen." The lag phase can last from a few hours to several days. Ideally, activity should start within 24 hours, but may take longer if a smaller population of yeast was pitched.

How to keep fermentation temperature "cool"?

Invest in a fermenter chiller that can maintain the fermenting temperature.

 

You can also put fermenter into a styrofoam box. Cut the lid to the size of the fermenter so that you can maintain the temperature of the fermentation within the box with the help of ice. The mineral water bottles can be used to freeze up ice and you can rotate these ice bottles about once every 8 to 12 hours. This will keep the fermentation temperature between 21 to 25 degree C. You can also buy blue ice pack from Diaso at $2 a piece. 

Is it really necessary to do a secondary fermentation?

No. But using a secondary fermenter allows you to rack beer off of the dead yeast and other precipitated solids. The resulting beer is cleaner tasting and less prone to haze or infection.

How long can I leave the fermented beer in the carboy without bottling?

The higher the beer's gravity the longer it may condition in the carboy. Most ales (SG: 1040-1050) can mature in a carboy for 1 to 1 ½ months. Strong ales (over SG: 1070) should be stored for a minimum of 4 months. Lagers should be aged longer than ales. The Brewer's Companion from Randy Mosher provides a useful chart on pg. 153.

My beer has bubbled up through the airlock; should I dump it?

No. Remove the airlock, thoroughly clean and sanitize. If the carboy is still foaming over, install a blow-off tube that fits snugly into the mouth of the carboy, with the other end submerged into a sanitized water solution.