Some good questions in here and I want to offer a programmer's perspective.

Does it help or hurt the game system to have the inventory display everything as "icons" instead of strictly by name like an old school RPG, or in what I really want to reference is the Grand Company Quartermaster menu.
Doesn't matter. From a purely technical standpoint the type of display is only relevant when you're looking at available screen real estate. Icons tend to be easier for players to parse visually rather than long lists, but they're bulky.. so if you're dealing with a lack of screen space, a list might be a better choice (FFXI uses lists for its inventory for instance... PS2 screen resolutions ahoy).

As it is now when we select an item we see the icon and it says the item name above the item, then we have the entire item help window to give the stats and ALSO show the icon for what it is.
Sure but bear in mind that the item tooltip (the big window that shows you the item's stats) is a UI element that can just be disabled. So printing the name of the item above it helps the user tell the item apart from something with the same icon when they're not displaying tooltips for whatever reason.

For the GC Quartermaster menu it is first separated in categories, Weapons / Armor / Materiel / Materials (lol) which seems nice and fitting. It then lists the item by name and shows a general item type icon that's much smaller. And again when selected it shows the actual item icon and stats in the item help box, this is obviously the same.
I suspect SE is separating long lists for things like vendors (GC vendor, tomestone vendor, whatever) into tabs like that because of the display requirements for all that information. So a vendor might be displaying not only icon and name, but also price with an interface to purchase multiples. The memory requirements for displaying all of that garbage at the same time get tricky especially when you're talking about the PS3 or a minimum-spec PC. I know SE dropped PS3 support, but I very seriously doubt they went back and redesigned every element that was sensitive to the PS3's base specs. And then for continuity, they've kept that same design moving forward because people are used to it.

Now I don't know about everyone else but I would find it much easier to find stuff if it was listed out by name easily to see in a list instead of just eyeballing it based on the icons until I select them.
Having two distinct display types like you're describing is a non-trivial amount of development and design time. Considering most people do better with icons than they do with lists (especially the enormous screen-devouring lists that our vendor system uses), icons are generally going to be a better choice for complex inventory systems.

But my question is more in a technical sense, do the icons help save data space / memory or would using a more text based listing actually help the inventory management in some way?
No difference, just visual.

And within each of those category tabs it could easily list 40 items given the above or more and you could just scroll through them down the list to find what you want and still tab over into other categories or switch "pages" by clicking where the ranks would be.
Just my opinion here but that sounds unbelievably cumbersome. I played FFXI for several years and dealt with their "lists of lists" inventory management. And it sucked. Finding things was a huge pain especially as they started adding more and more different types of inventory as the game got older.