Identifying numeric substrings of an alphanumeric string, and summing them
Learning to use, and using the debugger are fantastic ideas. It is a valuable skill that will serve you well. Understanding how your code works (and breaks) is very valuable, and the debugger helps you through there.

The remainder of your specific questions are somewhat linked to your actual implementation, and I am going to suggest that your implementation would be simpler with a String split operation. The specification says:


  Given a string, return the sum of the numbers appearing in the string,
  ignoring all other characters. A number is a series of 1 or more digit
  chars in a row.


Taking this literally, I would code this up as a Regular expresison system matching digits. Something like:

private static final Pattern NUMBERS = Pattern.compile("[0-9]+");

private static int digitSum(String s) {
    Matcher matcher = NUMBERS.matcher(s);
    int sum = 0;
    while (matcher.find()) {
        sum += Integer.parseInt(matcher.group());
    }
    return sum;
}


The Pattern says "Look in the string for a sequence of digits".

The method loops through all digit sequences, converts them to integers, and totals them.