Concatenating strings of equal length
You're not using StringBuilder effectively.  If you already have sb, then why not do sb.append(b).toString() instead of sb.toString() + b?  If you look at the generated bytecode for sb.toString() + b, you'll see that it actually makes another StringBuilder for you, like new StringBuilder(sb.toString()).append(b).toString().

I would write it this way instead, with no special cases:

public static String equalConcatentation(String a, String b) {
    int len = Math.min(a.length(), b.length());
    return a.substring(a.length() - len) +
           b.substring(b.length() - len);
}


I've chosen to make it a static method because it's purely a function of the two input strings, not using any object state.