I'm going to respond to something you weren't looking for - fiberglass batts on basement walls are not ideal. Yes - it is done all the time but foam is better. I'm not up on all the arguments but the basics are that moisture comes out of the concrete and will cause mold/mildew in the fiberglass. Far better to use rigid foamboard and I am pretty sure that you would be fine with R-8 - which would be 2 sheets thick. You can then put fiberglass on that if you want. Code here would be insulate the floor. Anytime you have unconditioned space next to conditioned, you need to insulate. Even if the basement is always going to be mild temp. I see what you are thinking - get some climate control from the ducting but that isn't allowed. And then maybe some more from the floor above. Fiberglass is cheap. Insulating the floor is the right answer. My builder always insulates the basement floor even on a conditioned walkout basement. The reasons are sound (minimal) and if you don't want to heat the basement at different times then your 1st floor won't suffer.

The wine storage will be cooler. If the area outside of the wine storage will not be high humidity, you don't need a vapor barrier. You want to stop air flow and insulate. Fiber bats will not stop air leaks. Spray foam will.

R 13 isn't really great insulation. In addition, the Whisperkool unit isn't designed to vent to the outside. Cooling units are designed to operate within a general range of temperatures. So a fridge, an AC, and the Whisperkool unit are all designed to operate within their own unique parameters. An AC unit usually goes 25 degrees or so and expects the top temp to be somewhere in the 90s. So you get cooling down to the mid sixties or so. They don't get things quite as cold when the exterior temps are in the high 90s or 100s. A fridge is designed to cool down to the mid 30s or so, and to operate in an ambient temp of around 70. That's way past the condensation point, so you have really tight seals and super insulation. The Whisperkool is designed to cool down to about 55 or so from an ambient temp of around 70, not the 90s you may experience outside. It's not rocket science tho. If you want to get cooler, you need a larger evaporation coil and a more powerful condenser and you help the machine with tighter seals and better insulation. Thus, you can leave your door slightly ajar with a window AC unit and you'll still get fairly cool, but you wouldn't leave the fridge door slightly open. To your main question - the one degree fluctuation is not all that bad. Don't sweat it. To your larger question that you didn't ask - you're possibly overtaxing the Whisperkool unit. If it dies on you, you can replace it with a regular window AC unit. Just seriously oversize it, then tweak the thermostat - either replace / bypass it, or tape a resistor onto the probe so that it thinks the air is actually warmer than it is and it cools a bit more as a result. It's how I cool my wine cellar.

what is the ambient temperature of the surrounding room? The assumption is that since it's in your basement, your floor above will be in a heated room - do you heat your basement too? Is your basement entirely subterranean or do you have 3 ft windows in it like mine? If you are entirely subterranean, and your ground temp is OK. you may not have to insulate the floor. Remember that it can be a heat sink and insulating it can actually raise the temp of the interior space. If your configuration is like I described, you need most insullation on the ceiling. The vibrations from your dishwasher won't be a problem anyhow - the dishwasher is above and it's hard to imagine how it would shake the floor of the wine cellar. I simply have the same concrete floor as the rest of the basement. My cellar is for storage, so I just wanted it to be functional. It's about the same size as you describe. Vapor barriers can be something to think about. If you have a large difference in temp between your cellar and the surrounding space, you can get condensation. in an unheated basement, the problem is more likely against the ceiling than the walls, but each place is different. Generally you put the vapor barrier against the warm side, but that's not a brilliant idea for a ceiling, especially if your joists and flooring is wood, which is probably the case. I don't understand your question about paint. What is your concern? Also, remember that greenboard doesn't really insulate all that well. I was worried about flooding in the basement even though we've never had any, so I used wonderboard, or cement board, like you would use for a bathroom. I also have six inch walls so I could put a lot of insullation in, and I built my own door which is also six inches. If you use extruded polystyrene, you get the maximum insulation per square inch. Thermax, which has foil on one or both sides, is slightly better but deteriorates over time and there is some literature about toxins, so rather than worry about any of that, I just used polystyrene. I also built my own wine racks. If you use individual racking, you lose a lot of space to air, but that is the most convenient for looking at your bottles. If you use some kind of bin, Rhone shaped bottles and anything that's slightly weird can be a real pain to stack. So I made shelves. A Bordeaux bottle is 3" diameter, a Burgundy bottle is about 3 1/4", but some of the Rhones are a little larger and things like Turley, or Champagne, or some Beaujolais, are even larger. So I made shelves spaced at 3 1/2 inches, with a few at the bottom that are 4" and the top which is about 5" for anything weird. That was the compromise that allowed me to get the maximum bottles with the least amount of lost space. Bins are better but as I said, a pain. I did one wall with bins and probably wouldn't do that again. For cooling, you have a lot of options. You can do a split system. You can get a Whisperkool or Breezeaire or something similar. But remember, those coolers are simply air conditioners. AC units don't go down below 65 or so because they'll ice up. So the compressors are sized slightly differently and the evaporation coils are sized slightly differently than the wine chiller units. If you got your AC to go to 60 or below, you wouldn't have adequate evaporation. So what to do? Get a larger unit. The smallest room AC is about 3000 BTU, or 5000 BTU, which is far more than you need to get that small room cool. If you trick it, you can get it to cool to the 50s or so and you're in good shape. I just bought a $99 Samsung and the temp is around 59 - 62 degrees F. You can pull the thermostat wire thru the grill, which gets you about 5 degrees right there, and you can tape a small resistor to it that gets you another few degrees, and you spent $99 for a cooling system that works fine. I'm not going to get into an engineering description of cooling systems, but people modify big ACs for meat lockers if they're hunters, and it actually works pretty good for a wine cellar. But a spare in case you burn it out and use the savings for more wine.