I've moved your second post here, as it seems to be the same issue and it's easier to keep all the information in the same place.
The Images node in the tree under your skeleton in Spine has a path. This is where it loads images from and images are relative to this path. It can be absolute, or relative to the project file. Eg, if the "char_player" folder is a sibling to your project file, you could set it to "char_player/standard" and it would load images from there. In that case your image name would be "player_std_back_pelvis". If you leave it blank or point it to the parent folder of the "char_player" folder, then the image name will be "char_player/standard/player_std_back_pelvis". Either way, this is just a name in the texture atlas. As long as your texture atlas look up uses the same name as the image name in Spine, it should work.
This doesn't help much, but if you do need to rename your images you can double click to rename. Another option would be to export to JSON, use find & replace on the JSON, then import back into the project.
However, I don't know there is actually a problem with 2D Toolkit loading an atlas that has image names the use a path. If there is, we could fix it in spine-tk2d, stripping off the path there. This works as long as you don't have two images in different subfolders with the same name.
I'm not a Unity guru and even less so for 2D Toolkit, but I'll try to provide as help as I can. Anything Spine related I can help for sure, but some of the more esoteric Unity stuff I'm not so familiar with yet. Also, Unity tends to be complicated with a lot of setup and pieces which isn't easily conveyed in a forum post. Any given problem could easily be in how you have configured your project, and that is difficult for anyone to know.
In your second screenshot there is no material for the mesh. In spine-unity the material comes from the atlas asset, I'm not sure why it doesn't show up for you in spine-tk2d. How is your atlas asset configured?
Have you tried taking the working spineboy example and replacing the assets with your own?