Q:

Drivetrain wear -- questions on chain elongation and skipping

Pitch (i.e. the distance between two links/two cogs) is a major issue in drivetrain wear, which explains why waiting too long to replace a chain may lead to premature cassette replacement.
This being said, three questions:
1) What is the maximum chain elongation tolerated by a new cassette? 1.5%? 3%? Can you explain in some amount of detail? (intuitively I would think that the answer involves taking into account the difference in sprockets' radius at the bottom and near the top of a cog -- as chain pitch increases, it rides higher and higher on the cogs, up to the point where it is no longer gripped by the cog).
2) What is the maximum chain elongation at which a chain will skip on sprockets that have not been changed during the life of the chain? (intuitively I would think that the answer would be the same as above, or a tad lower - worn sprockets have deformed cogs, but the same principles apply).
3) Large sprockets last longer than smaller ones. Why? (intuitively I would think that wear is a function of the number of revolutions -- or distance/speed)

A:

According to Park Tools:

A worn chain shifts poorly and wears sprockets at an accelerated rate.
  The CC-3.2 is a go/no-go gauge designed to accurately indicate when a
  chain reaches .5% and .75%, the points at which most chain
  manufacturers suggest replacement. For 9 and 10-speed chains, replace
  chain just as the gauge fits the 0.75% side fits flat into the chain.
  For 11 and 12-speed chain, replace as the 0.5% side fits. The CC-3.2
  is long, accurate and features permanent measurement markings.

The 0.5% - 0.75% answers when you should replace the chain but does not answer the max stretch before skipping.   
Sheldon Brown's site ( http://www.sheldonbrown.com/brandt/chain-care.html ) indicates things get dramatically worse once the elongation reaches 1.0% because at that point it is every other link that is load bearing... so I would assume >= 1.0% is the answer with the variance being attributed to the condition of the cog.  Checkout the link and you will learn more than you can imaging on chain-cog engagement.

The best way to determine whether a chain is worn is by measuring its length. A new half inch pitch chain will have a pin at exactly every
  half inch. As the pins and sleeves wear, this spacing increases,
  concentrating more load on the last tooth of engagement as the chain
  rolls off the sprocket, thus changing the tooth profile. When chain
  pitch grows over one half percent, it is time for a new chain. At one
  percent, sprocket wear progresses rapidly because this length change
  occurs only between pin and sleeve so that it is concentrated on every
  second pitch; the pitch of the inner link containing the rollers
  remaining constant. 

A:

Regarding your second question about exactly how worn a chain has to be before it skips on sprockets that are also worn, I don't believe that a nice clean engineering analysis is possible, because you would need a specific wear model for both the sprocket and the chain.  There are far too many variables: alloy composition, heat treatment, and surface finish of the various parts; size distribution, composition, and amount of dirt particles; how often the chain was lubricated; and so on.  Instead the industry has devised a practical rule-of-thumb measurement that works very well: the elongation percentage.  @dafew's answer describes that well.
On a personal note, let me add that chain skipping isn't the only danger of an elongated chain.  Once I had a BSO with a badly-worn chain.  I jumped on and started pedaling hard; the chain skipped and then caught, ripping several teeth off the cog, as I flew over the handlebars.  I ended up with some impressive bruises and road rash on my forearms.
As for why smaller cogs wear faster than larger cogs, consider a road bike with a 50-tooth front ring and an 11–32 cassette on the back.  On the 11, every turn of the crank will use each tooth 4.5 times.  On the 32, each turn of the crank will use each tooth 1.6 times.  The smaller cogs wear faster because they rotate more often.

