1. If I create different VLANs for different Departments in a switch, should I create INTERFACE VLAN for every VLAN created or can I simply Route that segment of IPs to a Core switch for Internet.
You need to have respected SVI Layer 3 interface each VLAN and it should be routing aware to reach each other.
to reach the internet if the IP is not public routable address, then you need NAT
2.If I enable IP routing in a switch, can I write Block specific VLANs from communicating (Lets say I create VLAN1, 2,3,4,5 I want to prevent vlan 2 & vlan3 from communicating and allow all other inter_vlan communication)
you can use ACL or VLAN ACL to achieve this.