Here's my lexicographic solution, which comes in at 89 characters (including the call to sort):

sub chipmunk_lexi {
sort{(($B)=$b=~($r='^\s*(0|-?[1-9]\d{0,8})(?!\d)'))<=>(($A)=$a=~$r)||$
+A<=>$B||$a cmp$b}@_
}
[download]

If I've understood the hole properly, the definitions of "integer" and "starting with an integer" are a bit tricky. '0' is an integer, but '00' isn't; '0a' starts with an integer, but '00a' doesn't. Thus, '0', '1', '00', 'adam' would be sorted in that order. Have I got that right?