Essentially there are many different designs you could go for, but they will largely depend on your needs, your IT staffs understanding of networking and budget.
You need to identify if you have any restrictions on who should be able to access who (for instance, should sales be allowed to communicate with PCs in accounting? Should certain servers only be accessible to certain groups?
Are you planning on using VOIP at any point in the foreseeable future (I would recommend you at least make provision for it)?
Do you have remote workers, VPNs etc?
When you say routers at each office for redundancy, do you mean a router at each office that connects to the internet?
Also, have you had it confirmed that interconnecting your offices with fibre is a ok. If so, will it be terminated in the patching/server/comms room?
Does your office currently have floor ports etc for user computers, that link back to a patching room?
If yes, is this where your servers are located?
What is the budget that you are trying to build this on?
In relation to your initial questions, I wouldn't recommend using 3rd party media converters like that. I would recommend using a Layer 3 switch with SFP adapter slots to interlink the buildings.
The number of switches depends entirely on the budget. If I were to build this for my company, I would be putting 2 48 ports in building 1 and another 48 ports in building 2. This gives plenty of room for expansion, as the cost difference between the 24 and 48 ports is much less than double.
Fibre trunking between switches is fine, but it could get expensive. Your professor is correct, at 400m, your only option is fibre if possible to install or using an ISP to implement a leased line or other WAN solution.
I know it's quite a lot to go through, but laying a good foundation to your network could save you huge amounts of money in the future, as well as preventing failures, security breaches and allowing for a scalable network.
Hope this helps.