Need to transform 900+ images to 600x600 and keeping their name.

Regarding the name problem...

Create a new folder, and save the image to that folder after resizing.  The action will then send all images to the new folder, and as the file does not exist in that location, it will save with the original name.  

Are you saving the resized images as TIFF ?  I don't know all the steps you'll be using, but if you are trying to save as JPG you might need to add Flattern Image as the last step in the action (before saving) or it might want to save as PSD or TIFF.

You can also use File > Script > Image Processor to batch resize. I don't think I am totally clear on what you are trying to achieve, but you might be able to avoid using an Action this way.  You can, or course, run an action in Image Processor.

Working on a copy/test file...

Use file/automate/fit image 600x600px. Stop recording.

Then undo or make the canvas anything but 600px in either widht or depth.

Start recording again. Then add a canvas size step to 600x600px (you'll need the canvas not to be 600 in width or depth to correctly record this step).

You may also need to consider a background colour or transparency depending on final file format and use.

All of this can be recorded into an action and used with the batch command or various image processing scripts.