I. is regular , all strings over {0,1} that start and ends with same symbol.
II. is also regular. we need to remove first symbol from stings accepted by L
Design DFA for L, Move with one transition from start state, and the states we reach, say some Q,mark them as new start state, it seems we have more than one start states, then add a new state as start state and connect it with these Q with null move.
Example
L = { all strings over {0,1} those start with 00 }, having regular expression 00(0+1)*
DFA for L
on giving symbols { 0,1} at start state we reach to q1 and q3, add new start state and connect it with q1 and q3 with ∊ moves.
After removing Null moves (∊-moves) , we have new DFA for Late(L)
having language Late(L) = { all strings over {0,1} those starts with 0 } having regular expression 0(0+1)*