that dryer duct thing, is it inside a black rectangular box? if so that is the air filter/air box.
As for the inline fuel filter, go to Wal-mart and go to the lawn and garden section, they have Briggs and Sratton tune up parts and there is one little "kit" that has a length of clear fuel line and an inline filter, they work perfectly.
The piece that broke off of the petcock, don't sweat it too much, there are three positions on a petcock, off-on-reserve, what happens in the on position is that the gas is pulled through the little meatl straw that was sticking up into the tank, the plastic thing on the end is a bit of a filter to stop large particles, you can live without it, when you go to reserve the gas is drawn from the base of the petcock. The point of the reserve is so when you run out of gas on "On" you can flip to reserve and get youar ass to the gas station.
Make sure you don't forget to mix oil with the gas, about 4 onces of oil per gallon of gas is good