1) $ L = \{ uww^{R}v : u,v,w \in \{a + b \}^{+} \} $
This is a regular language, If you look closely then you will see that we can stretch $u$ and $v$ such that we can get $w w^{R} $ as $aa$ or $bb$.
Associated Regular expression can be : $ (a + b)^{+} . a a . (a + b)^{+} + (a + b)^{+} . b b . (a + b)^{+} $
2) $ L = \{ uww^{R}v : u,v,w \in \{a + b \}^{+}, |u| >= |v| \} $
This is not a regular language, because you need a counter to check that if $ |u| >= |v| $ ?
Hence this is not regular.