Are we ever going to see quirks like these in a presidential election?

[The Arizonan]

Sometimes I look at the presidential election maps and notice interesting patterns. These make me wonder if these streaks are going to continue.

Are we ever going to see weird quirks in presidential elections like a Republican winning without Ohio and/or Arizona or a Democrat somehow losing Washington, D.C.?

[Don Vito Corleone]

Sometimes I look at the presidential election maps and notice interesting patterns. These make me wonder if these streaks are going to continue.

Are we ever going to see weird quirks in presidential elections like a Republican winning without Ohio and/or Arizona or a Democrat somehow losing Washington, D.C.?

Yes for the former, unless what it means to be a Democrat changes completely then no for the latter.
But really this is question without an answer, as you didn't really define "quirks", and you only gave 2 examples.

[The Chill Moderate Republican]

Are we ever going to see weird quirks in presidential elections like a Republican winning without Ohio and/or Arizona or a Democrat somehow losing Washington, D.C.?

I like that line. Maybe in a 100 years the republicans will start to win the D.C metro area and the democrats will lose support in much of the rust belt.

But on to your main point of this quote. Ohio last voted for the loser in 1960 to Richard Nixon and Washington D.C has always voted for the same party by huge margins since it was added to the electoral college. We can always guess what the electoral map might be, but the map will always change, just like the times. 

[Chunk Yogurt for President!]

DC might go Republican eventually.

-The parties could change on a lot of issues.
-The demographics of DC could change.
-The Democrats could nominate someone who is convicted of murder in the last weeks of the election.

Also, if the US capital is moved somewhere else, the government jobs would leave.

[SingingAnalyst]

"Quirks" are becoming less likely as party perceptions solidify, as they have in each of the last several (at least 5) Presidential elections.

Up until 2016, any rural, conservative, Southern or Appalachian county that voted Democratic (Knott, KY; Elliott, KY; and others) could be seen as a quirk. Alas, I am unaware of a single such "quirk" in 2016.

I suspect that in the 1950s, a Republican winning without NJ would have been seen as quirky, but it happened in 2000, 2004, and 2016.

A Republican winning without OH or AZ is certainly possible as the demographics of these states change. I don't see the GOP ever being competitive in DC at the Presidential level; the closest election I recall in DC is when Marion Barry defeated Carol Schwartz for mayor, 56%-42%.