Who decides when the passengers may board the aircraft? Is this a pilot or cabin crew decision, and what are the prerequisites / conditions (cabin checks completed, flight plan filed, ...?) for this decision?
It is a joint decision of the entire flight crew (flight crew consists of pilots and cabin crew).
It is mostly a decision by the most senior flight attendant because it is mostly a cabin issue. The prerequisite is, obviously, the cabin has to be cleaned and prepared. There are "cabin checks" to be carried out when the plane is empty. Typically, each flight attendant will be assigned an area. When everyone reports they're ready, then boarding can begin.
But there are factors from the cockpit that can delay the boarding process too. For example, ATC informs them that takeoff will be delayed for 2 hours. Or a maintenance is being carried out. So, the pilots would always been informed before boarding begins, but in most cases there'd be no objection.
Captain is always the person who makes this sorts of decisions. If the captain is absent, the successor of command decides.
There are some limitations, though.
a minimum number of cabin crew must be present (depends on aircraft type, roughly 1 per 50 passenger seats)
cabin checks completed
fueling not in progress
other company defined items completed (cleaning, catering etc)
For example in my company, pilots' presence is not always required.