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Do Google host their own products on Google Cloud, or are there different sets of data centres for Search/Drive/Gmail vs Google Cloud Customers?

This is a leased facility, the kind of place Google rents for cloud customers but doesn't need for itself. Google's own datacenters are https://www.google.com/about/datacenters/locations/

Other way around. Google Cloud runs on the same underlying datacenter, compute, and network infrastructure that Search/Drive/Gmail does.

[edit: at least in regions where Google HAS its own datacenters, e.g. "us-central-1? yes. europe-west9? maybe not"].

That does not imply that Search / Drive / Gmail runs on top of Google Cloud.