Motif, welcome to ArboristSite. Some bar oil dripping off of a saw is common and considered normal. Due to gravity, the bar oil which has been pumped onto the bar and chain while cutting will eventually make its way to the lowest point before dripping off. You might try momentarily opening the bar oil fill cap to vent off any built up pressure from within the oil tank when you are finished cutting. Another way of looking at it would be that the saw is not leaking. It is just marking its territory.
The oil residue collecting on the muffler is also a common occurrence. 2-stroke engines, by the nature of the design, are not extremely clean burning engines. A certain amount of un-burnt fuel mixture is expelled from the cylinder along with the exhaust. The type of 2-stroke oil and the amount mixed in the gasoline play a big part in this. A good quality synthetic oil mix will burn cleaner and help reduce the amount of residue collecting on the muffler.
It is more difficult to safely buck up shorter pieces of wood. There are several devices you can buy to help accomplish this more safely. Chain saw protective pants or chain saw chaps will help protect your legs from accidental contact with a moving chain. Steel toed work boots will help protect your feet. Good leather work gloves, hearing protection, and eye protection are all important. You want to keep the chain from coming in contact with the ground underneath the logs which you are cutting. Cut through most of the log, then roll it over and finish your cut. Some do a balancing act by keeping one leg on the log to stabilize it while cutting. It is not a really safe technique to use.
Here are a some tools for you to consider. These are available from Bailey's, whose customer service is legendary.
http://www.baileysonline.com/search.asp?SKW=TIMBERJACK HANDLE&catID=11507
http://www.baileysonline.com/itemdetail.asp?item=15730&catID=9730
http://www.baileysonline.com/itemdetail.asp?item=CB41&catID=9730
http://www.baileysonline.com/itemdetail.asp?item=SB1500&catID=9730