Operation on a regular language and context free language
There is no guarantee that your resulting language will be context free.

Here is a hint for creating a more formal proof.  Try the languages $L_1 = c^*$ and $L_2 = a^nb^n$.  The resulting language, given your additional requirement of $|u| = |v|$ would be $c^{2n}a^nb^n$.

Using the pumping lemma for context free languages, are there two substrings that you can pump indefinitely that will match this language?