First of all, there's a big difference between 開けるand the others
You use 開けるwhen someone or something opens an object intentionally.
彼はドアを開けた→He opened the door
あくand開くare almost interchangable and are used when something opened on its own usually unexpectedly.
ドアがあく→the door opened
ドアがひらく→the door opened
There are few words that only fit with either of the two so no need to worry