Learn Spanish (Latin America): Nadie puede abrir este candado viejo.

You only need a double negative if the negative word (like nadie) goes after the verb. Since nadie is placed before the verb here, it already makes the sentence negative, so you don't add no. If you wanted to put it at the end, you would say: No puede abrir este candado viejo nadie.

Why do we use the singular verb 'puede' for 'nadie'? Doesn't 'nobody' refer to all people not doing something?

In Spanish, nadie (nobody) and alguien (somebody) always take the third-person singular verb conjugation. You treat them exactly like you would treat él (he) or ella (she).

Why is it 'este candado' instead of 'esto candado'?

Este is the masculine singular word for 'this' when it comes directly before a noun, matching the masculine noun candado. You only use esto when you aren't naming the object, like if you just point at the padlock and ask, '¿Qué es esto?' (What is this?).

Could we put the adjective before the noun and say 'este viejo candado'?

It is grammatically possible, but descriptive adjectives like viejo usually go after the noun in Spanish (candado viejo). Putting it before the noun can make it sound poetic or emphasize your feelings about how old it is, rather than just stating the simple fact that it is an old padlock.