How to support the side of concrete steps?
I can't give a definitive answer for most of this as I have no building experience, but the answer I would give to no 2 removal of the excess concrete on your foundation is a very loud 'don't do it'. it is as you say part of your house foundation.  I don't think many would recommend reducing your foundation structure. If you wish this to be part of a social area, can I suggest that you find a way to increase this lip. And to use this as a shelf for cups, glasses  etc. You may even like to have the odd small plant pot on there. This can be done by tiling up the wall against the existing concrete, and then tiling the shelf, or by boarding up the concrete and having a wooden shelf.  Just a thought. 
How are you thinking about accessing your new patio area? Using the existing concrete steps may be awkward, and you may need to retain the soil pile and use this to have steps down to your patio, from the existing steps.