How I could create an auto-correcting text-editor application?

You tried to integrate Term::Complete into your curses UI, but that's not about auto-correction.

First thing you have to do is get the snippet of text which has to be examined as e.g. $candidate:

$cui->add_callback( 'editor', \&current_word);

my $candidate; # word to build a complete list from, or to spell-check

sub current_word {
    my ($x, $y);
    Curses::UI::TextEditor::getsyx($y, $x); # not $editor->getsyx($foo
+), this would overwrite $editor. BUG!
    $y--; # there's 1 padding line
    chomp (my $line = $editor->getline_at_ypos($y-1)); # first line is
+ 0
    my $part = substr $line, 0, $x;
    ($candidate) = $part =~ /\b(\w+)$/; # caveat locale, utf8
    warn $fh "x: $x y: $y '$line' - '$part' candidate: '$candidate'\n"
+;
}
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This will give you information to STDERR about the current word under the cursor at every keystroke.

You can build a list of completion candidates based on that word, or check the spelling with e.g. the aspell library.

Based on that word, you could decide wether adding a <Tab> to the text, or attempt completion.

You could mark misspelled words in terms of Term::ANSIColor or such, do completion with either Term::Complete or Term::ReadLine::Gnu (with both you have to shoehorn the results into your $editor widget) - or just do your own completion routine with Curses::UI using Curses::UI::Listbox to display possible completions.

PS: why? are you trying to re-implement vi or emacs in perl? ;)

PS2: "Find and Replace" comes first in functionality, then more sophisticated things like (auto-)completion and spellcheck.