A PID will learn your chamber's performance and narrow that range to a few tenths of a degree.
Probably not. Most systems use a refrigeration compressor with a minimum cycle time measured in minutes rather than in milliseconds. It can't feather the cooling to zero in on a temperature. A second PID controlling heat could manage that phase well but cooling won't be precisely controlled unless a different cooling scheme is used. Even then you'd have to figure bout how to make the two PIDs interact without constant "hunting".