If in your starting hand you don't get any basic
Pokemon but you do get a Mysterious Fossil, are you
required to use it as a Basic and send it out first or
can you draw seven new cards because Mysterious Fossil
is technically a Trainer card?  I have the same
question also for Clefairy Doll- does it count as a
Basic or a Trainer?

Since both Mysterious Fossil and Clefairy Doll are considered trainers in your hand and do not count as Pokemon until the are on the board, a hand consisting of only a Doll or a Fossil is considered a Mulligan hand.

Scott