Alice, I am a mason contractor.
A lot of your questions are dependent upon a bigger question; what is your budget like? You are DIY ing, so my assumption that you are looking to save money.
If I am correct, it only makes sense to use the existing foot print as much as possible.(id like to know more about the foundation material). This will save you a hassle of creating a new site, and you already have all the landscaping and utilities in place. I would hope that the original builder took exposure and view shed and other 'geomancy" into consideration when they chose the site in the first place.
If you go for a new foundation, I recommend concrete, unless you have a readily available large stone source, the time and effort it will take to make a full foundation from stone may be prohibitive, unless you live in an area where you can pull large blocks of limestone or sandstone that will shape uniformly and quickly. If you choose a new stone foundation, look in to slipform construction, works well for diy and can use locally available irregular stone.
You can still achieve a stone foundation look by applying a veneer to your concrete. Either use a brick ledge off your footing of your foundation, or install an angle iron build plate or you can use an adhesive mortar to apply a <1" thinstone. Another option for the "solid" look is to give your concrete a parge treatment, there are many techniques to decorate concrete to make it look more appealing. In this same vein, consider strawbale construction, it is great for insulation (cooling and heating) has a "solid" look when you apply stucco, and is relatively cheap and quick. It also works well for passive solar designs, although it appears you may not have the best exposure for that. Cob or dirt bag is also an option for you, but gather advice before you start as both of those are very labor intensive.
the other advice here is sound: spend a lot of time on your design and materials and stick with the plan, changing your mind mid build is a hassle for contractors, and gets really expensive fast.
Budget for building besides money also includes time: how soon would you like to complete your home? this could be a major factor when deciding what material you would like to use. If you are making your own lumber for example, plan on waiting a year for it to season. If you are planning on building a stone foundation yourself, research your techniques and ask a contractor about industry standards for production. My uncle (a building contractor) recently reminded me of the rule of 3/2: 3 times as long, and twice as much money to make a building as you initially (bid) expect.
If you have a healthy budget and you feel like you have money to throw at this project, I would recommend that you find an architect or experienced builder that you trust to help you with your design and logistics work. This is really important especially if you don't have a construction background. I can tell you stories of people who tried to build houses themselves without thinking things through and in the process ruined their finances, broke up families or at least caused themselves a big headache. Not to be all doom and gloom, I wish you the best of luck and encourage you on your adventure. Permies is a great place to ask advice, we are all here to help each other.