TOPIC:

Maybe something got wrong with your installation?!
How did you do the installation of JEM 2.2.1: Via the GUI of Joomla! or did you unzip the files and copy them onto your server manually?
Where is the file jem.css located?

As said. I use Joomla 3.7.4. This forum is for JEM 2.1.x on Joomla 3.4. Perhaps that is the problem.
Installed through the GUI of Joomla by uploading the package in the window.
The jem.css file is located as in earlier messages. All default done by the installer.

This topic is for JEM 2.1+ with Joomla 3.4+. Most of us use tha latest joomla . There are some prehistoric people who use Joomla 2.5.
So you are on the right place.
I am only a user, but I use clean installations and updated systems, but on all systems, I get these images, even with the links, I described in my posts.
EDIT:
Apropos:the jem.css of JEM 2.2.0 and earlier are on the same place with the same links and worked too!

I did a full update of a large 2.5 site. There are no errors reported.
The first file (icon-16-archive.png) and others are present in /domains/mydomain/public_html/administrator/templates/hathor/images/menu/

The links in jem.css do however not point to this folder. Something strange must going on if a request for these files in the stylsheet ends in an other folder with your installation. I have used Xenu (windows) to verify the new site, look for orphaned files and broken links. It's now 99,1% ok. Try this simple tool and see what it reports.

Hathor is installed with 3.5.1 and also with 3.7.4. So that is not the problem.
It is just that the link paths in jem.css should point to an other folder.
To: /administrator/templates/hathor/images/menu/icon-16-archive.png

So you are correct about the fact that these pictures are part of the standard Joomla install.

This should be the correct jem.css content, starting from line 1075:

images/admin/ doe not exist in the jem folder!

#jem .btn-micro i.icon-archive,
.jgrid span.archive {
background-image:url(/administrator/templates/hathor/images/menu/icon-16-archive.png);
}

#jem .btn-micro i.icon-trash,
.jgrid span.trash {
background-image:url(/administrator/templates/hathor/images/menu/icon-16-trash.png);
}

.jgrid span.default {
background-image:url(/administrator/templates/hathor/images/menu/icon-16-default.png);
}

.jgrid span.notdefault {
background-image:url(/administrator/templates/hathor/images/menu/icon-16-notdefault.png);
}

.jgrid span.checkedout {
background-image:url(/administrator/templates/hathor/images/admin/checked_out.png);
}

.jgrid span.downarrow {
background-image:url(/administrator/templates/hathor/images/admin/downarrow.png);
}

.jgrid span.downarrow_disabled {
background-image:url(/administrator/templates/hathor/images/admin/downarrow0.png);
background-position:0 -12px;
}

.jgrid span.uparrow {
background-image:url(/administrator/templates/hathor/images/admin/uparrow.png);
}

.jgrid span.uparrow_disabled {
background-image:url(/administrator/templates/hathor/images/admin/uparrow0.png);
background-position:0 -12px;
}

.jgrid span.published {
background-image:url(/administrator/templates/hathor/images/admin/publish_g.png);
}

.jgrid span.expired {
background-image:url(../images/publish_r.png);
}

.jgrid span.pending {
background-image:url(../images/publish_y.png);
}

.jgrid span.warning {
background-image:url(../images/publish_y.png);
}

.jgrid span.protected {
background-image:url(/administrator/templates/hathor/images/admin/icon-16-protected.png);
}

#jem .btn-micro span.icon-publish:before {
content: url(../images/tick.png);
}

#jem .btn-micro span.icon-unpublish:before {
content: url(../images/publish_x.png);
}

#jem .btn-micro span.icon-archive:before {
content: url(/administrator/templates/hathor/images/menu/icon-16-archive.png);
}

Thanks for your work. If there's another user who has problems to see these images in JEM, he can use these changes. (in my versions it works without these changes).