13
The LHS is
than the RHS. Here's how we're going to prove it.
OK, let's get started. We want to know whether $\frac23\left(\sqrt5-1\right)^3<\sqrt[3]2$. So,
That's the first step completed. Now
Nearly there. Finally,
(The above is fairly long, but only because I've gone into quite a lot of detail. The actual calculation is rather quick, and for those ...
9
I came up with this, though I have to admit I had to check the number of inhabitants of the last one:
4
With the new hints this has become much simpler. In fact, OP has (unintentionally?) changed the character of the puzzle. At least some solutions can be step-by-step (like a sudoku for want of a better simile) reconstructed from the hints.
For example number 8 (I'll only do one, so people still have a chance to earn the bounty by solving one of the others):
...
4
Well, I chose to solve this using Excel, which took me about 9-10 hours, I think - I suspect it would have been much quicker to infill a PDF image using MS Paint (or equivalent), or possibly even do it manually by printing it off! Still, I believe (unless I have made a silly mistake somewhere - and believe me, it's easy to do...) that the final solved mosaic ...
4
I used a depth-first-search program to find all possible chains of length 5. I used this list of cities: https://worldpopulationreview.com/world-cities
For a bit of fun I also wanted to find the longest chain possible. I only ran it for a few minutes and I am sure longer chains are possible. I found one with 57 cities:
3
Alas, I couldn't find a 5-city chain but I did find a 6-city chain.
Update: I found a 5-city chain but I tweaked a little bit. I used the old name for one of the cities:
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