Can you sell found weapons in Breath of the Wild

There is 1 unbreakable weapon in the game, but it has a caveat.

The Sword that Seals the Darkness (also known as the Master Sword) is effectively unbreakable. However, while using it, it will eventually "run out of charge" and can't be used for around 10 minutes, but it will remain in your inventory. The only exception to this is when the blade glows Blue. This means it is near enemies that the game considers to be greatly corrupted. While the blade is glowing blue, it has massively increased durability when striking these enemies. However, it will take normal durability damage when striking rocks/normal enemies/etc, even while glowing blue. It's damage also doubles from 30 to 60.

How to obtain it:

You must go to the large Woods shrouded in fog north of Hyrule Castle. Find your way through them (you cannot paraglider in) until you reach Korok Forest. Sitting in the middle of the forest will be the Master Sword. In order to pull it from the pedestal, you need 13 hearts, as pulling the sword out drains 12 and 3/4 hearts. Temporary hearts do not count for this, you need a full 13 red hearts.

There are a couple of other breakable weapons that are infinitely obtainable.

Each Divine Beast you defeat will reward you with a weapon that represents the Guardian who controls the beast. These can all be rebuilt by NPCs in the city where you obtained them when they break. They all require one generic weapon of the same type, 5 flint, and 1 diamond.

You can throw any weapon by holding down and releasing the upper right shoulder button (R). A thrown weapon will deal double it's base damage and stagger most enemies.

Source on double damage:

Critical Hits – Weapons that you have no use for are perfect for throwing at monsters. If one hits, it will deal double damage.

Keep in mind that throwing a weapon will take away a lot of it's durability, sometimes destroying it instantly.

Boomerangs fly back to you when thrown but you still have to catch them manually by pressing A in the correct moment.

Throwing a normal weapon is usually a good idea if there's only one hit left.