As a verb, เลย is to pass, to go beyond. Everything else it does comes out of that, but the three uses have drifted far enough apart that it is more honest to treat them as three words that happen to share a spelling and a thread.
They take different positions, which is what makes them separable in practice.
The three
After a negative: at all. It is an intensifier of the negation and nothing else.
After a verb or an instruction: right away, without further ado, go ahead. This is the one you will hear pointed at you across a counter.
Between a cause and its result: so, hence. Week 21 comes back to this one, because it turns out to be the ordinary way Thai builds a two-part sentence.
- ไม่ไปเลยmâj paj lɤːj
didn't go at all
- กินเลยkin lɤːj
go ahead and eat; eat it now
- ฝนตกเลยไม่ไปfǒn tòk lɤːj mâj paj
it rained so I didn't go
The thread
All three are passing beyond something: past the point where any at all would remain, past any delay before acting, past the cause and on to the result.
You do not need that to use them. It is worth having anyway, because it turns three separate memorisations into one word with three exits.