Is a Dedicated Interface, VLAN, and Self IP Required for a VIP in an F5 Configuration?

f5 is just like server.
it can have only 1 ip address to connect to many destination as long as having correct routing config in itself and next hop router.

if your f5 is HA pair, you need at least 3 ip for each configured subnet: self IP of F5A, self IP of F5B, and floating ip of f5 pair.

1. Is it correct that we need two Self IPs because the pool members are split across two different subnets?
No. With correct ip routing config, 1 floating ip can be adequate if it can connect to multiple subnets using this 1 ip.

2. Does the VIP require its own dedicated interface, VLAN, and Self IP?
No.
Just like server or even windows laptop , you can have 1 arm config that multiple VIP, self and floating IP of multiple subnets attached to 1 VLAN/1 interface.

3. The pool members, Self IP, and VIP cannot be in the same subnet as the management interface, correct?
Correct. control plane (mgmt) must be in different subnet than data plane (vips, selfs, floatings)