How can I safely remove the display assembly?

I would recommend using this article: MacBook Air 13" Mid 2013 Display Assembly Replacement

I have removed the clamshell many times and it does not damage anything when following these directions. I have not ever tried running the computer without the clamshell on it though.

Another option is to hook up an external keyboard, mouse and monitor and you can leave the clamshell closed and you will be able to use it as you are suggesting.

mmm.... You are between a rock and a hard place here ;-{

Did you try looking for a used display assembly locally? I would still try that first. a Mid '13 air still is a good system. Even if you've outgrown it you should be able to sell it to recoup the costs of the display and still have change left in your pocket ;-}

OK, thats not going to cut it here, so what to do? You really can't remove the lid without causing other issues. So how about a bit of surgery removing the LCD alone here (the glass) will that do? Follow the IFIXIT guide here: MacBook Air 13" Mid 2013 Display Assembly Replacement. That will allow you to gain access to the display for the next task pulling it out. Here is the next steps: A1369 13.3" Macbook Air LCD Screen Repair step by step guide. While Louis does this without taking the display out your damaged display will likely splinter as you take it apart. So taking it off first makes sense.