There is a key difference between these two constructions. I saw it + verb = We know that you saw the whole event/process See + verb=ing = We don't know if you saw the whole event/process For example, let's say that there was a full box of chocolates that was suddenly empty, and your parents want to know who ate them. You could say: 1. 'I saw Jane eating the chocolates' This means that you maybe walked into the room while Jane was eating the chocolates. All you know is that she ate some of them. 2. 'I saw Jane eat the chocolates' You saw the whole event, and you know that Jane ate all of the chocolates. In some cases the two forms are interchangeable, or it's not significant whether you saw all or just some of what happened - but the difference is there if you want to make it.

See it happen is usually used as part of a question. "Did you see it happen?"

Grammatically both are correct. It just depends on the context of what you are trying to express.

They are both fine, but it depends on the situation. 'happening' is present progressive tense. Let's say we're talking about a cat that knocked down a cup. 'Where did you see it happen?' 'I saw it happen in the store, the cat lives in the store and he is always knocking things down'. 'When did you see it happening?' 'I was shopping and I noticed it just when the cup was about to fall'. The difference in the second example is that the action is in the middle of occurring, it is still taking place, when you see it happening, but see it happen is present tense. So you could say "I see that happen all the time", or "I saw it happening and I didn't do anything to stop it". Sorry if my explanation is long, you should look at lots of examples. You can look here: http://corpus.byu.edu/coca/, search 'see' along with 'happen' and see the examples, then search 'see' and 'happening'. It takes some time to learn to use that website, so keep playing with it for a while and you'll get it.