Of the following 4 sentences, is 2 correct? [ . . .]
- Now that everyone is ready, let's get started.
- Now everyone is ready, let's get started.
- Now everyone is ready. Let's get started.
- Now everyone is ready, so let's get started.
All four sentences are correct. Like you, I don't like (2). To my native ears, the subordinating conjunction "now that" needs its "that." And in my experience, it almost always does.
Almost. Yes, sometimes "now" functions as a subordinating conjunction all by itself, and is used just as "now that" is used. I don't use it that way. I don't know anyone who does. But here are some quotes of the usage from the OED:
1816 J. Austen II. ix. 182 My mother will be so very happy to see her—and now we are such a nice party, she cannot refuse.
1889 ‘R. Boldrewood’ xli We'd as good as got a free pardon.., now the police was away.
1916 J. Buchan xiii. 174 Now you're in these pretty clothes you're the dead ringer of the brightest kind of American engineer.
1990 Dec. 36/1 (advt.) Now he's been in the wars himself, he says it's the RAF Benevolent Fund that really deserves a medal.
And here are a couple of quotes that I've come across in famous literature. In each of them, as with each of the above quotes, "now" can be replaced with "now that" without a change in meaning.
- "Now the shell was no longer a thing seen but not to be
touched, Ralph too became excited." (William Golding, Lord of the Flies)
- "And electric strength had come to him now the horizons were kicked out." (John Steinbeck, The Pearl)
As for your examples (3) and (4), they are also correct, as I mentioned earlier. However, in them, "now" is not a conjunction and cannot be substituted with "now that." In (3) and (4), "now" is simply an adverb: "Everyone is ready now."