Heh. Can't wait to see Railroad Tycoon's response to this...

IIRC, there are rules that allow a stopover (Tochuu Gessha / 途中下車) on journeys over 200km. For even longer journeys it's possible have more than one stopover. The rules are complicated...

The OP's journey from NGO to Nagoya will most likely be on a Meitetsu railway line, so that's going to require a separate ticket from the rest of the trip. Therefore, the JR part of this itinerary is likely to be a journey from Nagoya to KIX via Kyoto and Osaka. So I have a feeling that the idea of saving some pennies in this way is a non-starter. (Perhaps Nagoya to Osaka with a stop in Kyoto can be priced as a single ticket, but that doesn't have the potential to deliver huge savings)

I suspect it might be possible to issue a single ticket for a journey from Nagoya to KIX with validity over two days and one overnight stopover in, say, Kyoto. However, I doubt that JR will issue a single ticket for a Nagoya to KIX trip over more than two days with stopovers in Kyoto and Osaka.

The OP might want to consider whether it's actually necessary to stay a night in all those different places. Kyoto and Osaka are within comfortable commuting distance. Many people use one of those cities as their base to visit the entire Kansai region.

Instead of this "stopover" plan, I suspect that there are better savings by using a Kansai area pass (http://www.westjr.co.jp/english/travel/jrp/) to ride the Haruka to KIX.