Things couldn't have worked better on both of these questions.  Thanks to
all who responded to each, only one person had a suggestion for thematic
shading but it was enough to lead me in the right direction.

Issue #1: OVERFLOW ERRORS

Indeed the problem was that the FrontWindow() command returns a large
integer value that has nothing to do with how many tables are open.  So I
was really just using the wrong numbers.  I was able to circumnavigate this
problem by using the SearchPoint and SearchInfo commands.  The improved code
is listed below:

Sub Neighbor

Dim win_id As Integer
Dim foundobj, objtype, i, j, numlayers, numthemes As SmallInt
Dim  x, y As Float
Dim t_name, cstring, commandstring As String

win_id = FrontWindow()

If WindowInfo(win_id, WIN_INFO_TYPE) <> WIN_MAPPER Then
        Note "This tool only works on Map windows."
        Exit Sub
End If

x = CommandInfo(CMD_INFO_X)     'x coordinate of users click
y = CommandInfo(CMD_INFO_Y)     'y coordinate of users click

foundobj = SearchPoint(win_id, x, y)

If foundobj = 0 Then
    exit sub
End If

t_name = SearchInfo(foundObj, SEARCH_INFO_TABLE)

The select was rather straightforward with an obj within type statement

Issue #2: AUTOMATIC THEMATIC SHADING

This was a little trickier, but since nobody really suggested many things
I'll just paste the code in case anybody is interested.  Per John
Schlosser's suggestion, I used the unique values within one of the table
columns to give the shade command an idea of how to shade the map.  I ran a
for...next loop to create a long string containing all the individual values
in the column and then utilized Run Command to convert the string into a
shade command.

fetch first from neighbors                                      
cstring = """" + neighbors.Sector_name + ""","
for i = 1 to tableinfo(neighbors, TAB_INFO_NROWS)-2
fetch next from neighbors
cstring = cstring + """" + Neighbors.Sector_name + ""","
next
fetch next from neighbors
cstring = cstring + """" + Neighbors.Sector_name + """"

Run Command "Shade window " +win_id+ " selection with Sector_Name values " +
cstring


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