I will attempt to provide an alternative view on the questions.
You don't specify whether you are using ordinary Minecraft or MinecraftEDU, so this give me the chance to give a different view.
1. Yes you can have multiple worlds running if you are using ordinary Minecraft. I run a Bukkit Server
www.hullcraft.com and we run the
MultiWorld Plugin. This allows you to have multiple worlds on one server. ALthough Bukkit is dead and you will struggle to get an original Bukkit Server download you can still get a
Bukkit/Cauldron server here. Cauldron allows you to run both Server plugins (like on a Bukkit server) and Client side Mods (like on a Forge server). Best of both worlds.
2. Again, if they are using MinecraftEDU the comments from MisterA are very good advice. If you are running either Minecraft or MinecraftEDU you can also add any command to a list of banned commands in a separate file which will prevent the players form misusing the mods.
Again, running a Bukkit server where the customisation is held on the server and not on the client machine will still give you control over the players (worldGuard, PermissionsEX, AntiGrief, etc...) but will not put the load on the client machine. However you DO need to learn how to install these mods and customise them correctly (which is a great learning curve and one I always enjoy!)
3. Agree with MisterA on the MinecraftEDU choices. Normal Minecraft can run in offline or online mode. The good thing about this is that students can only log in if they have a genuine Minecraft account that they have paid for. The bad thing about this is that students can only log in if they have a genuine Minecraft account that they have paid for.....
Offline mode does not provide the recognition or authentication so is not good for the classroom. So for what you are looking for, MinecraftEDU would be a better choice for the classroom environment, but you could use Minecraft (on a bukkit server or Cauldron Server) to provide similar functionality.
4. Again, Agree with MisterA on the MCEDU stuff. A normal minecraft server on Bukkit etc... can also restrict access to blocks. You can create a full permissions tree using permissionsEX which will allow different levels of access to commands, blocks and other restrictions.
I give all of this information out as so often, I want the students to carry on working on the Minecraft world at home or open up the server to the public and let them in, so I try to find solutions that will work in both kinds of server.
There is no one solution that fits all (as I have spent many, many hours trying to discover) and sometimes MCEDU will be a better solution and other times a Bukkit or Vanilla MC server will be the best option.
The more you get used to setting up a server and customising it, taking it down, trying a different approach, the more comfortable you will be in getting different configurations to meet your needs.
Hope (some) of that helped...?
Joel