I'm not sure why 1.9.2 works for you - as far as I'm aware our servers should currently only work on 1.9 (not 1.9.1 or any newer versions). But Development may have updated the servers recently? I'm not sure. If the error pops up again...
All you need to do is change your minecraft profile to version 1.9 -- log in as usual to your minecraft launcher then in the lower left corner press the New Profile button. In the first field, enter a new "Profile Name" -- it should be something easy to remember (like 1.9) then on the Use Version field scroll to find version 1.9. Remember, don' use the 'Latest Version' or 1.9.1 or any newer version. Our servers will get there eventually - not sure when.
Here are some screenshots if you need them: https://help.mojang.com/customer/portal/articles/1475923
Afterwards, go to the server admin page and try connecting: http://mod.learntomod.com/server_instances/current There will be a steve tutorial video that will show you how to connect.
The my stuff < worlds page is just a record of worlds he's previously downloaded, backups in case of data loss. If he's downloaded the same world twice then those worlds will appear twice on the page). If he's trying to download a world and has forgotten where to download them -- click the 'what's hot tab' in the middle of the learntomod homepage then scroll down to worlds.
About the timed out error on the map - after about 10 minutes of inactivity in minecraft, the access will cut off and he'll need to press the start server button again to get a new ip and log back into miencraft. This is meant to be a new energy saving feature.