What is theoretical picture of late 19th century America that emerges from The Devl in the White City? How is that time like & unlike modern US?

I'm not sure what your use of the word "theoretical" means
in this context; however, I think perhaps you mean something like the stereotypical or
generic.  Anyway, that's the picture I'm going with.

The
turn of the century brought tremendous growth in America, and we especially see it in
The Devil in the White City.  This story is a little skewed for a
question like this because it takes place primarily in a big city--which was not the
norm for America at the time.  There are a few things we can determine about America
from the book, though.

1.  Big cities were dangerous and
corrupt.  Young girls came to town and were never seen again.  Graft and corruption were
omnipresent.  They were dirty and certainly works in progress.  In fact, the picture I
take from this reading is that Chicago (more than New York, which had been around longer
and didn't have the same character as a midwestern city) is a city trying to keep up
with its own growth.  It was a wilder town, because the people of the Midwest were,
shall we say, a little less refinedthan many Easterners--though New York had its share
of corruption, of course. It was just so easy for someone to get lost in the midst of
the busy-ness and unsanitary conditions and people taking advantage of naivete
(something Holmes took gruesome advantage of, of
course).

2.  People believed in the greatness of America. 
This was an event which drew millions of people; in fact, the numbers tell us nearly 30
million people attended the Fair from all over the world.  That number was almost half
the total population of the U.S. at the time.

3.  Americans
were creative and inventive and innovative.  Look at everything new which came from this
event--foods, energy, architecture, engineering, transportation, sanitation--you can
name plenty more, as you've read, too.

4.  Americans were
resourceful.  We see that over and over as Burnham and his crews had to overcome
obstacles--some of them overwhelming--time after time. 

5. 
This is an era in which America had "kings" of commerce and philanthropy.  This was
especially evident in New York, of course, as such "royalty" was just rising to power in
the West.  The word tycoon fits this era perfectly.

6. 
People (not just Americans) were still able to be awed by sights and sounds and
experiences which we would find rather mundane today.  The gondola rides, for
example, brought Venice to people who had generally only heardof
such wonders.  (Las Vegas has taken advantage of that same kind desire for experience,
but there is no real awe today.)  People were not in touch with the rest of the world
through every possible media, and travel was not as common for them.  Bringing sights
from around the world was awe-inspiring to the pioneering, manufacturing, working people
of America.

That should get you started,
anyway.