Router – Could someone explain why traceroute seems to be skipping a router on the network

You are right to not want to run multiple DHCP servers on the same network, that can end up giving very random results.

Right now it sounds like you have both wireless routers plugged into your main router by their internal ports, using them basically as access points instead of routers (just getting the wireless users connected into the main network and letting the main router do the real thinking).

If the wireless router the students are on allows you to setup your port blocking rules to specific addresses, you may be able to on that one set it to block all communication, then allow specific rules to your internal servers. Then connect that router to your main network by the Internet/WAN port on the router and enable DHCP on it. That will make it so that the students are segmented on their own network and can only access your main network routing through the wireless router, which has the rules setup on it.

So, you would have:
Internet Connection <-> Main Router WAN Port
Main Router Internal Port <-> Teacher Wireless Router Internal Port
Main Router Internal Port <-> Student Wireless Router WAN Port

Anyone connected to the main router or the teacher wireless will be in one network and have normal access. Anyone connected to the student wireless will be in another network behind the student wireless router and be subjected to the rules setup on that router.

Otherwise, you'd want to look into setting up a DMZ to separate the students from everyone else, then setup your specific rules they are allowed to do. Your current equipment may not support this, your main wired router would be the one to look at.

If all of the subnet can't go out, it is probably a firewall issue. Since you're using ASDM, you have an ASA firewall. Check the access list on the interface with 10.0.0.225 (assuming that is your ASA) and make sure you have ICMP and SSH allowed. Remember: ASA rules are automatically reflexive, meaning that any rule only needs to be set in the direction it is started (in your case from 10.0.0.0/24 to any, protocol ssh and ICMP). If you want to check whether or not the ASA should forward the packet, you can use packet tracer under tools in your ASDM. Two tips:

1. Make sure to select the correct starting interface (I always forget)
2. Make sure to uncheck Show Animation, since all it does is show an animation and make you wait longer.