What persists after a respawn in Minecraft?

No answer seems to get the full picture, so let's do a summary.

Here is what happens to your items after you die:

• Items can be picked up.
• When you die, all the items in your inventory are dropped on ground.
• Items on the ground are destroyed by explosions (creepers and TNT), lava and cactus.
• Water streams move items. Items float in water in 1.13+.
• Items on the ground get deleted after 5 minutes (time only counted when chunk is loaded).
• Items are affected by gravity and have no explosion resistance, health, etc. which other blocks/objects have. This means that even after they are dropped, a small explosion can kill all of them.
• Items placed in a chest are stored infinitely until removed, or until the chest is destroyed. The rules above then apply to the items dropped. Chests cannot be destroyed by fire, lava, or water, but can be destroyed with explosions.
• If you had any experience on you, a fraction of it will drop and be allowed to be picked up.

Keep in mind things don't happen such as mobs moving, creepers exploding etc. if the chunk is not loaded.

Your inventory is scattered around your death spot, and will stay there for approximately 5 minutes. Run for it!

To answer your three bullets:

• Is my inventory lost forever after death?

When you die all of your items scatter on the ground. On respawn you can go and pick them back up but they stay on the ground for a limited time. Unfortunately, if they disappeared then they are gone forever.

You can take advantage of this while raiding a dungeon, just charge in with a sword and upon dying run back to the dungeon, pick up your sword, and try again.

• Does storing items in chest help them persist after a respawn?

Items in a chest remain in-game forever, or as long as the chest persists. So anything you have stored in a chest will be there after a respawn.

• What happens to the contents of a chest if it is destroyed?

If a chest is destroyed its contents, and the chest block, will spill on the ground and can be picked up. If it's an explosion that destroys the chest, some of the spill will be lost.

Chests are not affected by fire.