Installation Possibilities? Suggestions?...

Nice looking house congrat's. With that open floor plan you should be able to put a good dent in your heating with a wood or pellet stove. Stoves are best thought of as space heaters so I would start with trying to figure out where you are spending most of your time and then see where you think it would fit best. Consider how you will get fuel to the stove so proximity to an exterior door is a benefit.

You can run a chimney on the outside of the house if you want to put a wood stove in the main part of the house but I think you may be right that it is better/cheaper/prettier to put it in the 1 story addition. Realize the chimney will need to clear of other roof-lines and obstructions as you make your decision.

Pellets are not cheap right now so many people who run on oil are struggling to decide if it's worth running the pellet stove given the price of oil. Don't know about gas. The downside of wood is you need to stack and dry it for 1-2 yrs before it will be ready to burn. The good news is that in a rural area and even many dense suburbs the fuel is often free if you don't mind doing some work. You'll have to make that decision for yourself.

Once you have selected a stove and a preliminary spot take a picture and post it saying what you want to do. Then people help choose a unit and can tell you what your install will need as far as clearance requirements and the chimney goes.

Yes you're on track. Take a picture of the area you're thinking of and what position i.e. flat against the wall or catty corner. An outside pic wouldn't hurt either.

Thinking pellet or wood? Have a square footage of the house, at least the level you want to put the stove in?