How to fasten one end of a rope to multiple points so as to distribute load equally?
You can use a concept from rock climbing here.  There are methods of attaching to multiple anchors that are meant to distribute the load to three anchors, such as a cordalette:  

 

You can search and find instructions for how to tie a cordalette, which might be all you need.  

There are other methods that are meant to have some slip so that the load is evenly distributed between the multiple anchor points.  Here is an example:  



You'll find lots of examples for this if you search for "climbing equalizing anchors" or something like that.