Your problem is that you assume that the repository URL is http://automapperhome.googlecode.com/svn/trunk/ automapperhome-read-only. The correct URL is http://automapperhome.googlecode.com/svn/trunk/. The automapperhome-read-only part in the SVN checkout command is the target directory, not part of the URL.
Update: Your other mistake is that you are using the SVN:// protocol for the checkout URL. You should be using the HTTP:// protocol. The only attempt you show in your question with the HTTP:// protocol is the first SlikSVN one, where you specify too many options; all other attempts use SVN://. Here's the (partial) output from SlikSVN on my machine:
C:\Users\francip\Desktop\Projects>svn checkout svn://automapperhome.googlecode.com/svn/trunk/ something
svn: Can't connect to host 'automapperhome.googlecode.com': A connection attempt failed because the connected party did not properly respond after a period of time, o
r established connection failed because connected host has failed to respond.
C:\Users\francip\Desktop\Projects>svn checkout http://automapperhome.googlecode.com/svn/trunk/ something
A something\tools
A something\tools\subversion
A something\tools\subversion\ssleay32.dll
A something\tools\subversion\license
The first command uses SVN:// and gets the same error that you get. The second one uses the correct HTTP:// and successfully checks out the source.
Update 2: You have to specify the target directory at the end of the checkout command. I was looking at the example command in your question and your comments and the only places I saw a target directory, it was in the form of c:development\automapper - a relative directory to the current working directory, which in your examples seems to be C:\Program Files\SlikSvn\bin - and this one usually is read-only, unless you are running as an administrator.
If that turns out not to be the problem either, I would suggest to remove all current outputs from your question, run svn checkout http://automapperhome.googlecode.com/svn/trunk/ %USERPROFILE%\Desktop\automapper and copy the command and the output verbatim from the console in the question. Baring more details, it's unlikely we will be able to help you further.
In any case, it seems unlikely that the problem is with your network configuration. The URL uses the standard HTTP protocol over port 80 and SVN is returning to you 200 OK, which indicates it's able to connect to the server. Whatever is going wrong is on the local side. Still, you could verify this by running Fiddler and trying again.