Learn Spanish (Latin America): Yo nunca saldría de este laboratorio.

Salir has an irregular stem in the conditional tense. Instead of using the full infinitive salir, you drop the i and add a d, creating the stem saldr-. Then you add the regular yo ending -ía. Other common verbs that use this 'd' trick include tener (tendría), poner (pondría), and venir (vendría).

Why do we use de here instead of just saying saldría este laboratorio?

In Spanish, the verb salir means 'to leave' in the sense of 'to exit' or 'to go out'. When you mention the specific place you are exiting, you must connect it with the preposition de (salir de = to leave from). If you wanted to say 'leave' without de, you would use the verb dejar (dejaría este laboratorio), which means to leave something behind.

Since nunca means 'never', do I need to add no before the verb to make it negative?

Not in this case. In Spanish, if a negative word like nunca comes before the verb, you do not use no. If you placed it after the verb, you would need a double negative: Yo no saldría nunca de este laboratorio. Both ways are correct and mean the exact same thing.