 Re: Omitting "that"
It depends on the grammatical function.
1. When it's a conjunction - ie in sentences like :
They said that they couldn't do it./ They said they couldn't do it
It's possible that he doesn't know about it. /It's possible he doesn't know about it.
then there is no "rule" - it's a matter of speaker choice. Generally, it's omitted in everyday spoken language, but is more likely to be included in more formal varieties of the language.
2. When it's a demonstrative adjective (Give me that book) or pronoun (I saw that!) it's obligatory.
3. When it functions as a defining relative pronoun, it follows the ordinary rules for relative pronouns - ie it can be used or omitted when in object position :
I saw the man that/who she talked to.
or I saw the man she talked to.
but is obligatory in subject position:
I need the train that/which goes to Oxford. |