I rechecked permissions and that was not the issue. The installer is trying to create the new file/directory in /httpdocs/components/com_jce which is already there.

When you said "No this is not normal procedure" does that mean I should not have changed it or JMS should have changed it for me?

NO.
If you are on the master website, and that the directory is already present, this is normal that you can not install JCE.
If you are on a slave site, the fact that the directory is present is not a problem because the content will be replaced.

Here, there problem is not the fact that the directory is already present but the fact that you can not copy a file probably due to a PERMISSION PROBLEM.
The message that you report me is very clear.
You can not copy a FILE and not a problem of directory.
Again, check the reason why you can not overwrite this file.
The usual reason is a permission problem.
Have you put permission chmod 777 on all the files to check this point ?
Have you checked the owner and group of the files and folders ?

JInstaller::install: Failed to copy file:
/var/www/vhosts/mysite.com/httpdocs/multisites/slavesite.com/tmp/com_jce_150/com_jce_150/site/index.html
to
/var/www/vhosts/mysite.com/httpdocs/components/com_jce/index.html

You are not allowed to copy the file "index.html" from the tmp directory into the com_jce directory.
List me the unix permission, owner and group associated to this com_jce directory and also the one of the file.
Tell me also which is the owner and group that use Joomla when you proceed to the installation of any component

I enabled ftp and it installed fine, just had to go back into master site and reapply patches. This leads me to another question:

How do I ensure security when ftp must be enabled to install components? PHP runs under Apache in my server setup, not CGI, which even though all folders/files are chmod 777 it prevents them from being created.

I don't know.
All I know is that this kind of problem is related to security and that when using FTP with a user that has enough permission, this work.
I know that permission problem may be due to owner / group problem and that when the owner and group is identical to the one used by apache, this does not cause any problem.
In general, the problem comes from the fact that Joomla has been installed (unzip) with a user that has more privileges than apache.