D

$\because DE$是$AC$的垂直平分线，
$\therefore AE=CE$，
$\because FE=CE$，
$\therefore EA=EF$，
故①正确，符合题意；
$\because EA=EF=CE$，
$\therefore \angle C=\angle CAE$，$\angle EAF=\angle AFC$，
$\because \angle C+\angle CAE+\angle EAF+\angle AFC=180^{\circ}$，
$\therefore \angle CAE+\angle EAF=90^{\circ}=\angle CAF$，
$\because AB=AC$，$\angle B=36^{\circ}$，
$\therefore \angle C=\angle B=36^{\circ}$，
$\therefore \angle CAB=180^{\circ}-36^{\circ}-36^{\circ}=108^{\circ}$，
$\therefore \angle FAB=\angle CAB-\angle CAF=18^{\circ}$，
$\therefore \angle B=2\angle FAB$，
故②正确，符合题意；
$\because AE=CE$，
$\therefore \angle C=\angle CAE=36^{\circ}$，
$\therefore \angle EAB=\angle CAB-\angle CAE=72^{\circ}$，
$\therefore \angle AEB=180^{\circ}-\angle B-\angle EAB=72^{\circ}=\angle EAB$，
$\therefore AB=BE$，
$\therefore AC=BE$，
故③正确，符合题意；
故选：$D$.