I will answer part of this question because i'm studying for a Macroeconomics exam...
Its a very complex answer but in short, for every dollar created, a dollar is destroyed. printing excess currency would create inflation, and lower the value of a dollar, which hurts the economy, target inflation rate, and the whole shibang.
In terms of value, each is a floating currency, meaning market prices determine the value. This is where the answer gets even longer, so in short, theres a money market, which determins how much demand there is for currency. This can't be controlled, only influenced by the central bank or federal reserve using the overnight interest rate, or treasury bills. This helps control the supply of money. The feds destroy old bills that the banks give them because they are torn or crumpled, etc. and want new notes created that are stronger.
Gold is one of the more stable commodities, so the price rarely fluccuates, as opposed to the dollar. In fact when there are times of trouble, people buy less "money" and more gold, which increases it's price. You understand supply and demand in its basic premise right?
The paper amount is guarenteed in a country due to the legality of it. It has to be accepted as currency in your country (only when it says "this note is legal tender").
Enjoy!