1 & 2: Somewhere in the Dakotas or eastern Montana?
For #1, one of the locations is in Eastern Montana. In 1988, Glasgow had the nation's high on 6/5/1988 with 108 and the nation's low on 11/17/1988 with -7 and 12/24/1988 with -6.
The other location is Elk City Idaho, but looking closer, I think the high is dubious. For 8/26/1986, the nation's high is listed as 106 at Elk City and on 10/18/1986 Elk City had the nation's low with 14. Looking closer, I don't think the 106 is valid though.
I would assume that in 1936, it could have happened in the Dakotas as well, but I don't know where or even if the daily nation high and low information can be obtained that far back (without going through all of the weather records by hand!).
2006 in South Dakota would be a good candidate as well, though I can't find any.
Edit: International Falls should almost certainly be on the list. It had the nation's high on 6/17/1995. It's a sure bet that it also had the nation's low sometime in 1995.
Two Harbors Minnesota also had the nation's high on 5/6/2016. It's possible that it may have had the nation's low the same year, but I'd have to go through the records.
For #2, the location isn't in the Dakotas or Eastern Montana.
Nope. Denver (surprisingly perhaps) has never dropped to 40 or below in July.
Yes. Denver has relatively mild winters, but they do sometimes have early and late season cold snaps. Even International Falls hasn't dropped below zero in as many months as Denver.
5) International Falls, I think?
Yes. International Falls has had three April days at or above 90. On 4/21/1980 it was 90; on 4/27/1952 it was 93; and on 4/28/1952 it was 91.
6) Either Milwaukee or Chicago.
Yes. I was thinking Milwaukee, but it could also be Chicago if you use the Midway Airport.
Here are the all time record lows for each location (at least at the airports):
Salt Lake City = -30 on 2/9/1933
Denver = -29 (some almanacs say -30, but don't give a date) on 1/9/1875
Milwaukee = -26 on 1/17/1982 and 2/3/1996
Chicago O'Hare = -27 on 1/20/1985
Chicago Midway = -25 on 1/20/1985 (some almanacs say -26, but don't give a date)
Coldest temperature thus far in 2018: -26 on 2/21
Warmest temperature thus far in 2018: 99 on 7/8 (All time record high)
Precip thus far in 2018: 11.01 inches
Snowfall thus far in 2018: 38.7 inches
Last frost of early summer: 7/1
First frost of late summer: 8/29
Last snow of late spring: 5/1
First snow of early fall: 10/6