7555 timer replacement
Yes, the NE555 will work fine in place of the 7555 CMOS version: There is nothing particularly low power being done in the circuit where the difference in performance between the two would matter. Your 9 Volt battery might last a bit less.
Yes, the 2n2222 will work in place of the BC108. Basically any general purpose small signal NPN transistor will work. The BJT is used in very low current condition, well within the voltage range of either BJT, and the circuit overall does not demand any particularly high frequencies.