Learn Finnish: Kyl mun ois pitäny ottaa taskulamppu ja sammutin! - Yes, I should have taken the flashlight and the fire extinguisher!

Ois pitäny is the very common spoken form of olisi pitänyt. This is the perfect conditional of the obligation verb pitää. In Finnish, you express 'should have' by combining the conditional form of olla (olisi) with the past participle of the obligation verb (pitänyt).

Why is the word for 'I' in the genitive case (mun)?

Whenever you use verbs of necessity like pitää or täytyy, the person who has to do the action must be in the genitive case. Mun is the spoken version of minun. The literal structure translates roughly to 'Of me it would have been required to take...'.

Why are the objects taskulamppu and sammutin in their basic nominative forms instead of taking an object ending like -n?

This is a classic rule for necessity sentences. When the main verb expresses obligation (like pitää), a total object does not take the usual genitive -n ending. It remains in the basic nominative case instead.

I recognize the start of sammutin from the verb sammuttaa (to turn off / extinguish). How is this noun formed?

You are spot on! The suffix -in is frequently added to verb stems to create a noun meaning a tool or instrument used for that action. So sammuttaa (to extinguish) naturally becomes sammutin (a fire extinguisher).