WxW^R is regular because of the fact that you can expand x to both sides as given x belongs to (a,b)*... now by expanding i mean to say;;
example- w=abb x=(a,b)*
wxwR= abbxbba as you know x is not defined so that we can expand x to both ends, in other words, x is treated as bbbb such that language looks like starting and ending with a. and DFA can accept it.
coming to wwwR it is not regular even it is not CFL because of ...let w=ab then wwwR= ab ab ba now it is impossible for DFA and PDA to store and compare even with a stack.