When is it not necessary to use "that"?
For example can I say "Do you think we are far enough?" Instead of "Do you think that we are far enough?"?
Are both valid?
If the first is valid, in which other situations can I "remove" the "that"?
Leaving off "that" is both more common and more idiomatic, but both are fine.
In your example, "that" is functioning as something called a complementizer, but that's something only linguists think about.
There are other cases where "that" is optional. For example,
- Here's a trick [that] I like to do.
- This is the car [that] I am going to buy.
- I have a book [that] I plan to read.
In these examples, "that" is a relative pronoun, and it can be omitted when it introduces a relative clause where it acts as the object, rather than the subject, of the clause. More linguistic babble.
There's also something linguists call WHIZ-deletion that allows you drop "that" in certain cases where it's followed by a form of the verb "to be." For example,
- This is a gift [that was] given to me by Mary.
- John has a car [that is] big enough for everyone.
I want to stress, though, that almost no native speaker knows these rules, and you shouldn't try to learn them. With enough practice, you'll get a feel for when "that" is required and when it's optional. In the meantime, it never hurts to leave it in.
Yes, both are acceptable in English. However, the first one is more common in the spoken language. There is one case where you cannot use "that". Here are four sentences:
He says he will leave soon. ✅
He says that he will leave soon. ✅
He says no. ✅
He says that no. ❎
The first 3 are correct, but the last one is wrong.
They're both valid, yes.
Apparently the "that" in that sentence is acting as a complementizer (something I'd never heard of before, since I've learned about 75% of my English grammar through other languages), so if you want to look up more information on it, that's the way to go. I found this discussion on it, which is kind of interesting.