No, almost certainly not. Unless you have political pressure in terms of one administrator effectively having access to everything, then stick to one domain. There are arguments with regard to the same DNS namespace being used, which might not suit a multi-branded multinational, but sounds like this isn't an issue for you. Again, this is all bollotics. In terms of scalability, AD now scales very well. Replication can be controlled quite nicely too. Things have moved on since Windows 2000 Server.
Flipping the question on its head, multiple domains increase operational overheads (from day-to-day user/group management, auditing, securing AD backups, proving recovery, Etc.), but also introduces the potential for configuration inconsistencies across domains.
Single domain... the way forward.
In terms of DC placement, don't get tooooo carried away with Microsoft's two-DCs per site model. Look at your WAN links, and more specifically, triangulation into sites. If you have redundant links, and the MTBF on those links is high, then don't over-engineer unnecessarily. I don't know how big / autonomous your schools are. However, if the latency on your links is high, then perhaps on-site DCs will be necessary. This whole argument comes down to the service level that your WAN gives you. You can always add additional DCs if required. Taking them away isn't quite as straightforward (experience vs theory).
Also, don't forget about Read Only Domain Controllers (RODCs), which work beautifully on server core. This might not be relevant for you, as it sounds like your schools are quite autonomous, but if you, for example, had a smaller school, which didn't / couldn't do its own user management, then an RODC would be fantastic.
In summary, get your bollotics nailed, then get a WAN survey sorted.