Merge and sorting arrays of String and int in more efficient methods
mergeAndSortStringArrays looks good to me. What comes to mergeAndSortIntArrays, you can clean it a bit:

public static int[] mergeAndSortIntArrays(int[] firstInt, int[] secondInt){
    List&lt;Integer&gt; merged = new ArrayList&lt;&gt;();
    IntStream.of(firstInt) .forEach(merged::add);
    IntStream.of(secondInt).forEach(merged::add);
    Collections.sort(merged);
    return merged.stream().mapToInt((Integer i) -&gt; (i)).toArray();
}


Generic implementation

Also, you could do 

public static &lt;E extends Comparable&lt;? super E&gt;&gt; E[]
mergeArrays(final E[] firstArray, final E[] secondArray) {
    final E[] ret = Arrays.copyOf(firstArray, 
                                  firstArray.length + secondArray.length);
    System.arraycopy(secondArray, 
                     0, 
                     ret, 
                     firstArray.length, 
                     secondArray.length);

    Arrays.sort(ret);
    return ret;
}


This, however, will not accept an int[] as the argument type, yet it will gladly work on Integer[] and other object arrays.

A minor tip

Instead of writing for loops for printing arrays, you can always say:

System.out.println(Arrays.toString(mergeArrays(first, second)));


Hope that helps.