About 100 items are in the Blog parent category under child categories - Health, Thoughts, Inspiration, and Science. I am trying to display the items in custom order, regardless of the sub category it is in (so the first item may be from sub category Health, the second from subcategory of Science). Is this possible?

You can select custom ordering ("ordering") in the "Item ordering" parameter of the category's settings and put the items you want manually to the top.

This is not working, actually. The item ordering when set to "Ordering" only allows the items to be ordered within their category. I have had to move all items into the same parent category in order to get the items to order correctly. As soon as you place an item into a parent category and try to place it in between two items of a different sub category it grays over. imgur.com/FAjnNiN

Sorry I misled you,
I meant featured ordering. You can mark items as featured and order them regardless of their category.

Featured items will appear on top, and the rest will follow.
There are also other options (Featured First etc) which you can select, to see which one fits your needs better.