As I talked about in a previous post, there were many other ways to do business in Ancient Rome without dealing in physical coin. All markets had a banker sitting at a table or bench in the middle of it. Goods bought at the market could be deducted from your bank account and deposited into the account of the vendor. Or the vendor could put things on your tab, to be settled later, if you both trusted each other. Large transactions too, like property sales, were often mediated through accounts, without any physical coin exchange. So coins were not needed in a significant portion of daily life.
When Romans did use coins, the Denarius (silver) was used most often. It varied in purity over time, but even pure silver is relatively strong. It’s not like today’s paper money that gets folded, wet, ripped, etc and needs to be reprinted. So while I don’t know what the average lifespan of a Roman coin, unless people were putting coins in sacks to use as weapons, the coins should have lasted practically forever. Contemporary findings of Ancient Roman coins show the majority of them to be more or less perfect.
I don’t remember reading anything about what happened to money when a new ruler took power and put his face on new coins. But the same system of bronze / silver / gold coins was constant, so changing money was as simple as melting down the old coin and minting new coin with the same metal. Coins were divided into increasingly smaller denominations over time, but that wouldn't matter for your old coin. Not all bankers were sanctioned assayers or minters, but it wouldn’t be hard to find one in any decently sized city. Bankers existed at many different levels and of different positions on the social ladder. There are gravestones that tell us that there were even slaves of the Imperial family who were deposit bankers and minters.
During the Roman Empire, the silver content of the Denarius went through steady decline. It was about 97% pure during the time of Augustus (27 BC - 14 AD), and down to 20-25% under Gallienus (253 - 268). You can see a chart showing the decline of the silver content of the denarius over time here (also shows rough values of inflation): http://wiki.mises.org/wiki/Money_and_banking_in_Ancient_Rome#Inflation_in_numbers
So any coin you had with the previous emperor’s face on it would still be worth at least its full value if it was still in your pocket (ignoring inflation). But obviously when the silver content became increasingly low, newly minted coin was increasingly worthless from the start.
http://penelope.uchicago.edu/Thayer/E/Roman/Texts/secondary/SMIGRA%2a/Argentarii.html
Banking and Business in the Roman World, Jean Andreau, 1999