Can this new laptop supports my old RAM DDR3 1600mhz (came from my old Lenovo G480 Laptop)?

If they are both DDR3L types, yes, it should work. However, if the old ram is just DDR3, without that"L", it may not. The "L" means low voltage type of ram, which can work at lower voltages (as well as higher). However, the the motherboard only supplies the "L" voltages (lower), regular DDR3 ram will not run properly in it.