This is a tricky question. The answer is: 1) and 2) are the same but 3) is something totally different. In other words, Don was a bit sloppy when writing "single character" and "a symbol". In both cases he meant
<math symbol> -> <character> | <math character>
as documented in BNF notation on page 289 in the TeXbook.
To understand the behavior and the examples one has to look hard into code and/or in various places other in the documentation, e.g., tex.web. Key to understand your examples are the pages 290-291 in the TeXbook. Here Don says:
A <math field> is used to specify the nucleus, superscript, or
subscript of an atom. When a <math field> is a <math symbol>, the
$f$ and~$a$ numbers of that symbol go into the atomic field. Otherwise
the <math field> begins with a {, which causes \TeX\ to enter a
new level of grouping and to begin a new math list; the ensuing <math
mode material> is terminated by a }, at which point the group ends
and the resulting math list goes into the atomic field. If the math
list turns out to be simply a single Ord atom without subscripts or
superscripts, or an Acc whose nucleus is an Ord, the enclosing braces
are effectively removed.
We can observe this behavior with the following small little document:
\documentclass{article}
\begin{document}
$
A
{A}
{BC}
\hat A
{\hat A}
{\hat {BC}}
\showoutput\showlists
$
\stop
On the terminal we then get:
### math mode entered at line 3
\mathord
.\fam1 A
\mathord
.\fam1 A
\mathord
.\mathord
..\fam1 B
.\mathord
..\fam1 C
\accent\fam0 ^
.\fam1 A
\accent\fam0 ^
.\fam1 A
\accent\fam0 ^
.\mathord
..\fam1 B
.\mathord
..\fam1 C
The A and the {A} both produce the same: a single \mathord atom with its nucleus being \fam 1 A. But {BC} is different, you see additional ord nodes inside.
The other three lines show that the accent "migrates" out of the braces if it is the only atom inside. This what is meant on page 291 and answers question 3):
A single Acc is an Acc atom by itself, e.g., \hat{BC} is one, even though its nucleus has 2 math symbols inside.
So now for creating math accents or math operators: your example seems to indicate that "a single character" or "a symbol" refer only to Ord atoms. But this is not true, the reason that your example behaves as it behaves is due to the above definition of the <math field> behavior.
If we examine you example by removing the \hat on each line, e.g.,
$
{\mathchar"0141}
{\mathchar"1141}
{\mathchar"2141}
{\mathchar"3141}
{\mathchar"4141}
{\mathchar"5141}
{\mathchar"6141}
\showoutput\showlists
we get
### math mode entered at line 3
\mathord
.\fam1 A
\mathord
.\mathop
..\fam1 A
\mathord
.\mathbin
..\fam1 A
\mathord
.\mathrel
..\fam1 A
\mathord
.\mathopen
..\fam1 A
\mathord
.\mathclose
..\fam1 A
\mathord
.\mathpunct
..\fam1 A
and we can see that except for the first line the outer \mathord (which is our mathfield) not just contains \fam1 A but another atom, ie a two-level structure not a single symbol.
So if that is becoming the nucleus of an Acc atom it is first turned into a box possibly with some spacing attached inside and then the accent is placed and that accounts for the difference.
To prove my point, let's remove the braces in your example, which gives very unreadable but perfectly valid code (as a single <math symbol> is one possibility for <math field> and \mathchar is such a symbol, see BNF notation on page 289):
$
\hat\mathchar"0141
\hat\mathchar"1141
\hat\mathchar"2141
\hat\mathchar"3141
\hat\mathchar"4141
\hat\mathchar"5141
\hat\mathchar"6141
\showoutput\showlists
And now we suddenly see that indeed any <math symbol> is possible and there is no difference as long as it isn't part of a bigger sub-formula in braces.
### math mode entered at line 3
\accent\fam0 ^
.\fam1 A
\accent\fam0 ^
.\fam1 A
\accent\fam0 ^
.\fam1 A
\accent\fam0 ^
.\fam1 A
\accent\fam0 ^
.\fam1 A
\accent\fam0 ^
.\fam1 A
\accent\fam0 ^
.\fam1 A
And of course the same is true if you try to produce a \mathop. The issue is that the moment you use {...} the stuff inside becomes a complex structure (a math list), even if you put only a single atom inside, unless --- and that is the exception documented in the paragraph above from the TeXbook --- that single atom is an Ord atom or an Acc atom.
(Deep breath) ... feels a bit like TeXbook p125 :-)
Best Answer
I don't know the package you mention and you didn't provide an example but using
\mathoparound the construction will revert to the operator limits positioning.