Demonstrative Adjectives, When can't we omit "that"?

Hi people!

I know of some cases when "that" can be omitted, as in: "I told him (that) he was looking great tonight."

Can we always omit "that" when it has this function? Here, I see it as merely connecting the two clauses.

BTW, which part of speech is "that" in my example above? Is it a subordinating conjunction?

Now, I know that, in some other cases, we can't omit it. Why is this so and which are these cases? (Of course I'm not speaking here about demonstrative pronouns.)

Now that I come to think of it, I believe this is an example of a sentence where "that" can't be omitted, but of course "that" is here a different part of speech (relative pronoun): "The dress that is lying on the bed is mine.", "The table that is broken looks awful in the room."

And, again, a case where I think it can't be omited and functions as a subordinating conjunction:

1. That your brother is a bore is undeniable.

What about this other example, which is very similar, but I doubt whether we could omit "that" here or not:

2. It is true that dental work is expensive. (I wouldn't omit it here, but I'm not sure.)

Now, what about these cases? Can "that" be omited here?:

3. I was sure that she was right

4. The belief that rates will rise soon is making everyone panic.

Here, "that" is functioning as a complement of an adjective (1) and of a noun (2).

Lastly,

5. All that glitters is not gold. "That" is the subject here, isn't it? I think it is almost obvious it can't be omitted but what part of speech is it? A relative pronoun as well?

Thanks a lot!

Mara.