Everyone else did most of the brilliant deduction and have some insights into the hints that got us here (e.g. conversion mistake) but here is what may be closer to a final answer?
You are the
First, beg for the most positive among the stables
Now you can do the second, beg for the independent.
Now you can do the third, beg for the overlappable.
Now you ...
Partial Answer (mostly complete except for the subtle&moderate hints): following the hints by Jens and Stiv, the three are (it also explains Decisive Hint 4):
I will explain the 7 things in order, and talk about the hints when they become relevant in the discussion.
The seven things referred to in the text are
and their symbols are ...
[Edited to reflect changes made in the question]
The completed crossword:
This time, Gladys visited
There were three things unclear to me, but those were answered by OP in a comment below this answer (and for the third thing, OP edited the question). Anyway, here were my initial questions:
To follow on from @Randal'Thor's findings and our discussions in comments, I believe I have now sussed that the mystery country the OP is seeking is:
To find this, firstly note that Exhibits 1, 2 and 3 are supposed to be 'equivalent' (as per Hint 1). We have already sussed that we are supposed to extract specific letters from the name of the countries and ...
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:
Identifying the countries
An important realisation, along with recognising maps, is that
Exhibit 1 - two parts
Exhibit 2 - four parts
Exhibit 3 - one part
Exhibit 4 - twelve parts
(thanks to @Stiv for some help here!)
The OP says that "Exhibits 1, 2, and 3 demonstrate equivalency", so maybe we're meant to get:
Then the meaning of ...