There are cool season grasses (Bluegrass) that come up early, thrive in cool humid weather, go dormant in the heat of summer (unless doused with large volumes of water and provided some shade) and then come back in the fall. These do well here, so long as they have irrigation and partial shade, however they are more likely to suffer during a drought.

Then there are warm season grasses, that don't come out of dormancy until mid-spring and back into dormancy once the cool weather sets in. The benefit of these grasses is that they thrive during our dry years, but at the same time they don't look great in the spring (especially since weeds can outcompete the dormant grass) or fall.

Given the taxing effect on water resources of everyone watering lawns during the summer droughts, there has been a push by the extension and government for homeowners to be more responsible with water resources promoting warm season/drought tolerant grasses. There is definitely an upside but some drawbacks as I mentioned above. 

We are at the extreme of extreme north for Zoysia grass however, and SDSU or U of Minnesota have limited resources on whether lawns will do well with this type of grass since it will suffer winter dieback. That said, your lawn looks really good! I would attribute the dead patches to winter salts and/or dogs using that patch since its at the end of the walk way, even just piling snow in that one spot could cause dieback. You might want to pull up that section with a trowal to just make sure it isn't grubs to cover all your bases.

Buffalograss is an alternative that is advertised for the Dakotas. However, even if you showed me a picture, I have limited experience with these particular grasses and couldn't tell the difference. 

Additionally, I have heard from someone who grew buffalograss that mixing warm and cool season grasses isn't a great alternative since they just alternate between dormant patches so you get a checkered green brown lawn! 

I've also just read researching your problem that these warm season grasses need less nutrients, so the typical 2-3 nitrogen applications may have a negative consequence and will definitely benefit the weeds. 

This is what I found from U of Missouri on Buffalograss...it had this helpful tidbit at the end and would probably apply to zoysia:

https://extension.missouri.edu/publications/g6730 

"Buffalograss does not compete well with weeds because of its low density, so stands of buffalograss tend to be weedy. Annual weeds may be controlled with spring applications of preemergence herbicides labeled for use on buffalograss, such as Dimension, Ronstar G or Surflan. Other preemergence herbicides may give equally effective annual weed control. Broadleaf weeds, including dandelions and plaintains, may be controlled with any of several products that contain 2,4-D and are labeled for buffalograss. For control of winter annual weeds and invasive cool-season grasses, such as tall fescue or Kentucky bluegrass, spray Roundup on dormant buffalograss in winter before spring greenup begins."

I also dug up this old article from the star-tribune...granted its 2007 but makes the same points others have:  https://www.startribune.com/just-say-no-to-zoysiagrass/11301591/#:~:text=However%2C%20planting%20zoysia%20grass%20in,warm%2Dseason%20grasses%20%2D%2D%20St.
 

Ok, so lets address the specific quesitons:

Now, I am wondering, was my neighbor right? ... maybe possibly. I would assume so, or it could be buffalograss - but looking at photos of buffalograss it definitely looks more like something out of the prairie compared to what you have. 

If so, how do I care for it given our cooler climate?  Also, I have a lawn service that fertilizes and controls the dandelions ... you have a good approach going, since warm season grasses don't handle weed pressure really well, attacking the weed early and again in the fall are the best times. 1 year of weeds = 10 years of seeds so definitely don't let any of them go to flower.

I don’t have sprinklers, so I don’t water often, if ever. Is that a problem? ... this is exactly why buffalograss and zoysia grasses are being advertised! 

I bag my clippings, but should I? ... I never say no to mulching. Think about it, you just mowed up all your fertilizer and tossed it out. Mulch-mowing is one free fertilizer application! plus, the broken down grasses add a fine layer of mulch which will keep additional weed seeds from germinating. Also, infrequent mowing, a nice tall stand will keep down weed pressure and need less water and be less affected by drought stress.

Would aeration or other lawn services help? ... yes, let the company know your grass type. I doubt they will do anything different, but the owner may possibly have more info than I do. An aeration cant hurt, though this grass is only growing in summer, so a fall aeration means the gaps in sod and plugs will remain visible all fall if that is an issue for you. 

Where do I find zoysia grass seed in South Dakota to replace some brown patches? Good question, I don't know who is selling it local, I'm guessing it is mail ordered.

Thank you for the great questions. I took me some time to research and I've never addressed this type of lawn question. 

I would also suggest you propose these questions at the next garden hour webinar:

https://extension.sdstate.edu/event/garden-hour-webinar-8

John Ball, the extension Forestry Specialist is the person I've gotten the most information on Buffalograss from, and he is in Sioux Falls all the time so he may have additional information on this subject. Tuesday, 7pm CDT!