You can remove both entries in known_hosts. They'll get re-populated when you connect to other servers.

There is an obvious error in the third command you typed. You need to put your pub key inside your home directory's ssh folder, and it needs to be in a specific file. If the school's server is using openSSH, you should add your key to ~/.ssh/authorized_keys . If the school's server is using a commercial ssh, then you may have to convert your pub key to their format.

Anyway, let's start from the top. You said that, if you remove all the contents from your .ssh folder, you can ssh to the schools server. So, go ahead and do that, and try to log in and log out the server a couple times just to make sure your login and password work. If you wiped everything off your .ssh directory, you should get a message asking you if you want to accept the server's public key, which should populate .ssh/known_hosts with one entry. While you are there, check to see if you have a .ssh directory inside your home directory, if you don't create one, give it 400 permissions

Code:

$ mkdir .ssh
$ chmod 400 .ssh

After you have established that you can ssh to the server and log in with your username and password, go ahead and create your key. The command below includes the -N flag. If you don't want to supply a passphrase every session, then -N "" will set your passphrase to nothing.

Code:

 ssh-keygen -t rsa -N ""

Now, you'll copy your id_rsa.pub to your home directory in the remote server. Again, assuming the school's server is using openSSH, you'll put your public key in authorized_keys.

Code:

scp ./ssh/id_rsa.pub remote.server.edu:~/.ssh/authorized_keys

Now you should be able to ssh from your computer to the server. If you can't, then we'll need to know the error you received, and take it from there.

About StrictHostKeyChecking=no :
What this does, basically it allows the connection to continue even if you get back the error you posted, that the server pub key and what's in your known_hosts file doesn't match. It is possible that the school server's administrators change the key frequently for whatever reason, and that is why they instruct to add that line to your ssh config. I have that entry in my lab servers, but I don't know that I want to use it in production.