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 $\frac23\left(\sqrt5-1\right)^3<\sqrt2$. 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 ...
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):
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 ...
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: