Hmmm..... tricky.
hiraku -- you can hiraku a door, but I think 'aku' is more common. You would use hiraku to open a bank account.
aku vs. akeru is just transitive vs intransitive.
mado ga aku (the window opens)
mado wo akeru (open a window)
"aku/aiteiru" should be a big part of anyone's vocabulary.
seki ga aiteiru (seat is empty)
omise ga aiteiru (the store is empty/not busy)
doa ga aiteiru (the door is open)
hajimeru vs hajimaru is also tranasitive vs intransitive.
kaishi suru... err.... it pretty much means hajimeru. I'm looking at example sentences in my dictionary but I think you could swap kaishi with hajimeru for all of them. :/
HTH