Use of こと を, と, こと が etc. to join sentences

Hi,

I'm writing because I don't find a rule to understand clearly how does it work to join two different sentences with verbs like "can", "Say", "think", "remember" and so on.

Here are the sentencesI'm working on:

Sentence 1:

"Do you remember what she said?": 彼女が言った こと を 覚えていますか

In this case it is used こと を to join the two sentences.

Sentence 2:

"I said that I understood": 私は分かりました と 言った

In this case it is used only と to join the two sentences, and not こと を like in the sentence before.

Sentence 3:

"I can't believe what he says": 彼の言う こと は 信じられない

This case is still different: it is used こと は to join the two sentences.

Sentence 4:

"Can you dance?": 踊る こと が できますか

Here it's used こと が; I think that it's the same than use こと は, or to say better it's the same use of は and が like in simple sentences.

So, is there a grammar rule to join two sentence, or does it depend by which particle holds the verb?

Thanks to whoever will help me.