Let me try to make this clear...

Synaptic and all package managers put the downloads in one place:
/var/cache/apt/archives/. You can see them in any file manager or from
the command line, however, the names are a bit cryptic. If you click
on them they will probably install. If you remember what you searched
for in Synaptic you can go back there and do the same search again. If
it is installed there will be a green block beside the name. If it is
red you can't install it because of something missing or other errors.

Before you close Synaptic look at the properties (right click) and on
the third tab, "Installed Files" you will see everything that Synaptic
put into your system to make that item work. If it is a non-menu,
command line, app (and I try to avoid them myself!) then one of those
installed files will be the name of the main executable. You can then
start the app by typing that into a terminal window - you might need
to be root (sudo before the name).

If the box before the name is blank then for some reason the
application didn't install. Got to Synaptics preferences on the menu
bar and check the second last box: "Apply changes in a terminal
window". Try again and watch carefully for any error messages.

Can you give us an example of something that you installed that did
not show up on the menu? If you find this happening often then you may
be an adventurous soul who is trying to install something that is not
what you hoped it would be. Or you just may have been unlucky. There
are many items in the repositories which are obscure and useful only
to people with advanced skills and peculiar interests. You probably
don't want them...