Does and M3 handle well? YES
Like any other driving situation the car only handles as well as the driver behind the wheel. I have a buddy who can make anything handle well, even a 1971 Valliant with a column shift.
Autocross... look the M3 has DSC. I can tell you that the few times my M3 was out in the snow and ice the DSC saved my ass. With it off it proved to be very fun and very dangerous. If you're going to beat her through rough roads then I would say no as the repair bills will get up there. If it's for a track car then you can't go wrong. Keep in mind the M3 is heavy but has good power and feels light in high speed cornering. Just make sure your rear shock mounts have been replaced with the stronger reinforced ones. Engine maintenance will also hurt the wallet. The car isn't fragile but it will require a lot of TLC which =$$$.
"If everything seems under control, you're just not going fast enough."
- Mario Andretti
1979 320is RiP
1994 318is Sold
1995 325is Sold
1995 530i Sold