D

$\because AD$平分$\angle BAC$，$DE\bot AB$于$E$，$DF\bot AC$，
又$\therefore DE=DF$，
$\therefore$结论①正确；
在$Rt\triangle BDE$和$Rt\triangle CDF$中，$\left\{\begin{array}{l}{BD=CD}\\{DE=DF}\end{array}\right.$，
$\therefore Rt\triangle BDE$≌$Rt\triangle CDF\left(HL\right)$，
$\therefore BE=CF$，
$\therefore$结论②正确；
$\because Rt\triangle BDE$≌$Rt\triangle CDF$，
$\therefore \angle EBD=\angle C$，
又$\because \angle ABD+\angle DBE=180^{\circ}$，
$\therefore \angle ABD+\angle C=180^{\circ}$，
$\therefore$结论③正确；
在$Rt\triangle ADE$和$Rt\triangle ADF$中，$\left\{\begin{array}{l}{AD=AD}\\{DE=DF}\end{array}\right.$，
$\therefore Rt\triangle ADE$≌$Rt\triangle ADF\left(HL\right)$，
$\therefore AE=AF$，
又$\because BE=CF$，$AE=AB+BE$，
$\therefore AB+AC=AE+AF=2AE$，
$\therefore$结论④错误，
故选：$D$.