ICSE Class 10 string programs, written the way the board marks them.

import java.util.Scanner;
class PalinWords {
    public static void main(String args[]) {
        Scanner sc = new Scanner(System.in);
        System.out.println("Enter a sentence in capitals:");
        String s = sc.nextLine().trim() + " ";   // a space at the end closes the last word
        String w = "";                           // the word being built
        int count = 0;
        for (int i = 0; i < s.length(); i++) {
            char ch = s.charAt(i);
            if (ch != ' ') {
                w = w + ch;
            } else if (w.length() > 0) {
                String rev = "";
                for (int j = w.length() - 1; j >= 0; j--)
                    rev = rev + w.charAt(j);
                if (w.equals(rev)) {
                    System.out.print(w + " ");
                    count++;
                }
                w = "";                          // start the next word
            }
        }
        System.out.println();
        System.out.println("Number of palindrome words = " + count);
    }
}

equals is used, not ==, because two Strings with the same letters can be different objects.

import java.util.Scanner;
class PigLatinLine {
    // returns the Pig Latin form of one word in capitals
    static String pig(String w) {
        for (int i = 0; i < w.length(); i++) {
            char ch = w.charAt(i);
            if ("AEIOU".indexOf(ch) >= 0)
                return w.substring(i) + w.substring(0, i) + "AY";
        }
        return w + "AY";                         // no vowel: just add AY
    }
    public static void main(String args[]) {
        Scanner sc = new Scanner(System.in);
        System.out.println("Enter a sentence:");
        String s = sc.nextLine().trim().toUpperCase() + " ";
        String w = "", out = "";
        for (int i = 0; i < s.length(); i++) {
            char ch = s.charAt(i);
            if (ch != ' ')
                w = w + ch;
            else if (w.length() > 0) {
                out = out + pig(w) + " ";
                w = "";
            }
        }
        System.out.println(out.trim());
    }
}

"AEIOU".indexOf(ch) >= 0 is a one-line vowel test: it is -1 for any consonant.

import java.util.Scanner;
class ShiftCode {
    public static void main(String args[]) {
        Scanner sc = new Scanner(System.in);
        System.out.println("Enter a message:");
        String s = sc.nextLine();
        String code = "";
        for (int i = 0; i < s.length(); i++) {
            char ch = s.charAt(i);
            if (Character.isUpperCase(ch))
                ch = (char) ('A' + (ch - 'A' + 2) % 26);
            else if (Character.isLowerCase(ch))
                ch = (char) ('a' + (ch - 'a' + 2) % 26);
            code = code + ch;                    // digits, spaces, signs unchanged
        }
        System.out.println("Coded: " + code);
    }
}

ch - 'A' gives the letter's place, 0 to 25. Adding 2 and taking % 26 keeps it in range, so Y becomes A and Z becomes B.

import java.util.Scanner;
class DoubleLetters {
    public static void main(String args[]) {
        Scanner sc = new Scanner(System.in);
        System.out.println("Enter a sentence in capitals:");
        String s = sc.nextLine().trim() + " ";
        int pairs = 0;
        String w = "", big = "";
        for (int i = 0; i < s.length(); i++) {
            char ch = s.charAt(i);
            if (i < s.length() - 1 && ch != ' ' && ch == s.charAt(i + 1))
                pairs++;                         // same letter twice in a row
            if (ch != ' ')
                w = w + ch;
            else {
                if (w.length() > big.length())
                    big = w;                     // keeps the first longest word
                w = "";
            }
        }
        System.out.println("Double letter pairs = " + pairs);
        System.out.println("Longest word = " + big + " (" + big.length() + " letters)");
    }
}

The test i < s.length() - 1 comes first, so charAt(i + 1) can never run past the end of the string.