Hexagolf: Validagons

R, 184 bytes

Golfed, could probably be golfed by a few bytes

function(m){e=min;f=max;l=length;v=which(m!=" ",T);a=v[,1];n=l(v[a==1,2]);u=(v[a==e(a),2]);all(u==v[a==f(a),2])&all(c(e(d<-v[a==ceiling(f(v[,1])/2),2]),f(d))==c(u[1]-n+1,u[l(u)]+n-1))}

Ungolfed, very messy, more like half way golfed

f=function(m) {
  v = which(m!=" ",T)
  a = v[,1]
  n = length(v[a==1,2])
  u=(v[a==min(a),2])
  c1 = all(u==v[a==max(a),2])
  d = v[a==ceiling(max(v[,1])/2),2]
  c2 = all(c(min(d), max(d))==c(u[1]-n+1,u[length(u)]+n-1))
  c1 & c2
}

Since the input format is unspecified, Input needs to be specified in an R array format, looking something like this.

         [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
[1,] " "  " "  "5"  " "  "6"  " "  "7"  " "  " " 
[2,] " "  "8"  " "  "9"  " "  "0"  " "  "1"  " " 
[3,] "2"  " "  "3"  " "  "4"  " "  "5"  " "  "6" 
[4,] " "  "7"  " "  "8"  " "  "9"  " "  "0"  " " 
[5,] " "  " "  "1"  " "  "2"  " "  "3"  " "  " " 

Here is a generator function that generates the input. The generator doesn't generate an object that is fed into hexagon check function, but rather the code specifying the array (effectively the same thing). So this should not count as parsing the text. Remember that I'm not inputting text, but rather an array structure.

generate = function(x) {
  s = strsplit(strsplit(x, "\n")[[1]], "")
  m = max(sapply(s, length))
  dput(do.call(rbind, lapply(s, function(x) c(x, rep(" ", m-length(x))))))
}

For example, the generated code would be: structure(c(" ", " ", "2", " ", " ", " ", "8", " ", "7", " ", "5", " ", "3", " ", "1", " ", "9", " ", "8", " ", "6", " ", "4", " ", "2", " ", "0", " ", "9", " ", "7", " ", "5", " ", "3", " ", "1", " ", "0", " ", " ", " ", "6", " ", " "), .Dim = c(5L, 9L )) which is identical to array(c(" ", " ", "2", " ", " ", " ", "8", " ", "7", " ", "5", " ", "3", " ", "1", " ", "9", " ", "8", " ", "6", " ", "4", " ", "2", " ", "0", " ", "9", " ", "7", " ", "5", " ", "3", " ", "1", " ", "0", " ", " ", " ", "6", " ", " "), dim = c(5, 9))

Hopefully this input method is in compliance with the rules.

Here are the test cases

x1 = 
"  5 6 7
 8 9 0 1
2 3 4 5 6
 7 8 9 0
  1 2 3"

x2 =
" # _
+ + +
 9 :"

x3 = 
"    t h i s
   i       s
  a         h
 e           x
  a         g
   o       n
    ! ! ! !"

x4 ="    5 6 7
   8 9 0 1
  2 3 4 5 6
   7 8 9 0
    1 2 3"

x5 = "r e c t a
n g l e s"

x6 = "  h e l l o
  w o r l d s
t h i s i s b
 e t a d e c
  a y n o w"

x7 ="  *
 * *
* * *"

x8 ="   .....
  .......
.........
  .......
   ....."

Generate input arrays

sapply(mget(paste("x", 1:8, sep = "")), generate)

Test for hexagon

sapply(.Last.value , f)

   x1    x2    x3    x4    x5    x6    x7    x8 
 TRUE  TRUE  TRUE  TRUE FALSE FALSE FALSE FALSE