Verified answer

Your examples are correct, but here is where it gets confusing. If you were to say, "I'm having the breakfast." it could refer to a specific breakfast that you have planned out, like for a company or something. Also, if you said, "Breakfast is very delicious." without saying 'The', it could mean you think breakfast tends to be delicious regardless of what kind of food you are having.

Note that in your particular case, I would say 'it' instead of repeating 'the breakfast' in the second sentence. So: "I am having breakfast; it's very delicious."

You don't really even have to state that's it's breakfast again since they already know what you're referring to.

just say,

I am having breakfast. It is very delicious.

I am having breakfast.

It is delicious.

Breakfast is delicious.

The breakfast that you served me is delicious. (Which breakfast? The specific breakfast you served me)

Yes it is correct, you have two separate sentences.

You can have one sentence and use an article:

I am having breakfast, it is very delicious.