Permission with いい and negative invitations with ませんか

いい just means "good" or "fine" and isn't a grammatical term in itself, but it's used often in patterns about permission. After て form it’s asking for permission to do the action marked by て. てもいい also works mostly the same and you’ll see either quite often.

食べて(も)いい？
Can I eat?
いいよ
Sure

When people use よかった they're normally expressing gladness, not goodness. よかった shouldn't be used for permission.

来てくれてよかった
I am glad you came.

When negation is involved, "permission" statements with いい often walk the line between permission and request, just like excessively polite English. When it comes to negative + ていい, both ないで(も)いい and なくて(も)いい mean roughly the same thing and are mostly interchangeable. なくて(も)いい is more common.

食べなくて(も)いいよ
It’s okay if you don’t eat

As a response, いい(です) usually assumes the meaning of rejection (like “I’m good” in English).

レジ袋ご利用ですか?
Do you need a bag?
いや、いいです
Nah, I’m good.

In a previous lesson we saw the volitional form can be used to invite/urge someone to do something with you. There is another way to ask someone if they want to do something together with you, and that is to use a question phrased in the negative. This is similar to the English “won’t you… ?”.

明日、映画を見に行かない？
Why don’t we go see a movie tomorrow?

This structure can also be used for requests that aren’t invitations.

すいません、もう一回言ってくれませんか？
I’m sorry, could you repeat that again? (lit. “won’t you say it again?”)