NAT/PAT is giving a boundary device a public IP and then its “Children” private IP’s…
It's more the other way around. You create a network using private IP addresses (for lack of public addresses) and then use NAPT as a means to enable communication with the Internet or any other network that can't make use of your private address range.
For outbound traffic, you translate your meaningless addresses to a public address (or one from a pool) that makes sense to the destination on your network edge. For inbound traffic you reverse the translation.
Whether the nodes are all in the same network or building or if the are distributed around the globe (e.g. using VPN or other tunneling) doesn't really matter. What matters is whether the far node can use the original source address (then don't use NAPT) or not (do use NAPT).
So it's more a matter of "administrative distance" rather than of geographical distance.
Core 1,2&3. Can they all use the same NAT/PAT Scheme? Would Core 2 understand the same NAT/PAT address as Core 1 has implemented?
Likely, but not necessarily. It depends on the overall address design.
By the way, the 11.55.150.x addresses you're using in the diagram are no private addresses as defined in RFC 1918.
Would Core 4 have the same NAT/PAT scheme as 1,2&3 or would it be completely separate/fresh?
Yes, they could use the same, contiguous private address space. Connecting them requires tunneling across public address space/the Internet.
Would Core 4 be on a different Vlan or a different private network completely?
VLANs are a concept for data link layer virtualization. That has (almost) nothing to do with network/transport layer translations.
A L3 subnet often lives within a VLAN, usually in a 1:1 relationship. For routing between VLANs you'd either have a contiguous addressing scheme and use normal, transparent routing, or different addressing schemes, requiring routing with NAT/NAPT.