Can you tell if a system is set up for load balancing from a domain name? Surely the domain name could resolve to a machine/load balancing system/shitty home router
You can often gather enough information to make a reasonable guess here. DNS load balancing is easily detected.
In the more common case of a load balancing appliance:
Does the number of domains which resolve to the same IP exceed what is likely for virtual hosts on a single box? Not conclusive, good indication. Does it set common load balancing, sticky session cookies like BigIP, AWSELB, etc? It's either load balanced or someone is trolling you. When requests are repeated, do headers switch between a few static values, or if sequential values are included, can they be separated into more than one sequence? Good indication. At a certain point you have to start getting tricky with traffic analysis deeper into the packet, where success depends on how intrusive the appliance is.
EDIT: This isn't just telling 'from a domain', but given a domain you can take further steps to fingerprint things.
From the sounds of it, I doubt these guys did any of that, given that they wanted to balance the load between one server. Or were they mad because they presumed load would exceed one endpoint? What was their objective in unblocking IPs?