The first problem I see in the screenshots is that in #3, you've got a "*deleted*" folder under your "My Library" folder. That's a problem because what I see in that *deleted* folder is what should be in the "root" folder of a mapped DAZ Studio library. When you try to load that sweater, DAZ Studio looks for the "data" folder directly under the root of the library the product is in. But in this case, that library is "My Library," so it's going to look for My Library/data/, but on your drive the files it's looking for are in My Library/*deleted*/data.
If you want to separate out stuff by source (i.e., having that separate *deleted* folder; I do this myself), I'd suggest first moving that entire *deleted* folder somewhere else, outside of My Library, and then adding it as a mapped library: Click that menu button up above all the tabs for "Install," Content Library," "Smart Content," etc., and select "Content Directory Manager." Under "Content Sets" on the right select "DAZ Studio Formats", which will enable the Directory "Add" button on the left. Click that, and just select that root *deleted* folder (wherever you moved it to). You can even add the same root folder under "Poser Formats," if you've got any products in Poser form in there that you want to be available.
Regarding getting Install Manager to put stuff on your storage drive, how did you set that up? What I did was to go into the Install Manager Advanced Settings, on the Installation tab. In the bottom part of that screen is a table of "Label" and "Path". You can edit an existing one, or add a new one (the + button at the bottom), and then select the one you want Install Manager to use in the Content Path Shortcuts drop-down. I don't know if there's an easier way, but that let me set up an install location on my external drive, and tell Install Manager to use it by default; you'll also need to make sure that directory is added in the Content Directory Manager as described above, so DAZ Studio knows where it is.