answer for question 1.
YES, each VALN should have different gateway.
on layer3 switch configure vlan2 interface IP as 10.96.100.1, vlan 3 10.96.101.1 & vlan 4 as 10.96.102.1.
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answer for question 2.
you can configure DHCP server on layer3 switch itself, by the following.
ip dhcp pool department1
network 10.96.100.0 255.255.255.0
dns-server
domain-name test.com
default-router 10.96.100.1
ip dhcp pool department2
network 10.96.101.0 255.255.255.0
dns-server
domain-name test.com
default-router 10.96.101.1
ip dhcp pool department3
network 10.96.102.0 255.255.255.0
dns-server
domain-name test.com
default-router 10.96.102.1
ip dhcp-server 10.96.100.1
ip dhcp-server 10.96.101.1
ip dhcp-server 10.96.102.1
Note: on layer2 switch, specify the vlan on the ports, based on that, the pc will pick up the ip address from the respective DHCP pool.
hope this helps,
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