What are the conditions for the Hardmode bosses to spawn naturally? When do Hardmode boss summoning items drop?

Solution 1:

The Destroyer

Night time is the only condition required for summoning. It can also spawn naturally as of 1.2 and spawns following the prompt 'You feel a quaking from deep underground...' or 'You feel vibrations from deep below...'.

The Twins

They have a chance of spawning when night falls once Hard Mode is enabled, displaying the message "This is going to be a terrible night..." at dusk.

Skeleton Prime

He can be summoned with the Mechanical Skull or has a very small chance of spawning at a random time. His approach is signalled by "You feel the air getting colder around you" or "The air is getting colder around you".

Plantera

After at least one of either The Destroyer, Skeletron Prime, or The Twins has been defeated, a pink bulb will appear somewhere in the Underground Jungle. Destroying this bulb, which requires a Pickaxe or a Drill will summon Plantera.

Golem

It is summoned by using a Lihzahrd Power Cell on the Lihzahrd Altar, which is found in the Lihzahrd.

quoted from the Terraria Wiki

All regular-mode bosses can still be summoned in Hardmode.

As you see, there's a tiny chance of every Hardmode boss spawning at night.

Solution 2:

Each hardmode boss has a chance to spawn every night (unless another boss is spawning that night) if that particular boss has not been defeated in the current world. Similarly, all enemies will have a 0.1% chance to drop the summoning item for a hardmode boss that has not yet been defeated in that world.