(1)解：$\because BD$是線(xiàn)段$AE$的垂直平分線(xiàn)，
$\therefore AB = BE，$$AD = DE.$
$\because \triangle ABC$的周長(zhǎng)為$18，$$\triangle DEC$的周長(zhǎng)為$6，$
$\therefore AB + BE + EC + CD + AD = 18，$$CD + EC + DE = CD + EC + AD = 6，$
$\therefore AB + BE = 18 - 6 = 12，$
$\therefore AB = 6.$
(2)解：$\because \angle ABC = 30^{\circ}，$$\angle C = 45^{\circ}，$
$\therefore \angle BAC = 180^{\circ}-30^{\circ}-45^{\circ}= 105^{\circ}.$
在$\triangle BAD$和$\triangle BED$中，
$\begin{cases}BD = BD \\DA = DE \\AB = BE\end{cases}$
$\therefore \triangle BAD\cong\triangle BED(SSS)，$
$\therefore \angle BED = \angle BAC = 105^{\circ}，$
$\therefore \angle CDE = \angle BED - \angle C = 105^{\circ}-45^{\circ}= 60^{\circ}.$
