I don't recall any of the characters in Lord of the Rings or The Hobbit calling Sméagol Gollum in his face. It seems to be more of a private epithet. Is this correct? Does Gollum know that others call him Gollum?
ROTK - No mentions of him being addressed as Gollum (I searched entire book). As per @poke's comment, he was referred to as "Sméagol" instead.
Fellowship - obviously no, as we don't see anyone talking to Gollum yet.
The Two Towers - Yes. Here are a all the instances I found:
But Frodo sprang up, and drew Sting from its sheath. With his left hand he drew back Gollum's head by his thin lank hair, stretching his long neck, and forcing his pale venomous eyes to stare up at the sky.
'Let go! Gollum,' he said. 'This is Sting. You have seen it before once upon a time. Let go, or you'll feel it this time! I'll cut your throat.'
'No, we won't,' said Frodo. 'But we won't let you go, either. You're full of wickedness and mischief, Gollum. You will have to come with us, that's all, while we keep an eye on you. But you must help us, if you can. One good turn deserves another.'
Sam got out the rope. 'And where were you off to in the cold hard lands, Mr. Gollum?' he growled. 'We wonders, aye, we wonders. To find some of your orc-friends, I warrant. You nasty treacherous creature. It's round your neck this rope ought to go, and a tight noose too.'
'Well, Gollum, or whatever it is we're to call you,' he said, 'now for it! The Moon's gone, and the night's going. We'd better start.'
He took a few steps away and looked back inquiringly, like a dog inviting them for a walk. 'Wait a bit, Gollum!' cried Sam. 'Not too far ahead now! I'm going to be at your tail, and I've got the rope handy
At last Sam could bear it no longer. 'What's all this, Gollum?' he said in a whisper. 'These lights? They're all round us now. Are we trapped? Who are they?' ... 'Come, Mr. Frodo!' said Sam. 'Don't look at them! Gollum says we mustn't. Let's keep up with him and get out of this cursed place as quick as we can – if we can!'
'I don't like the sound of it at all,' said Sam. 'Sounds too easy at any rate in the telling. If that path is still there, it'll be guarded too. Wasn't it guarded, Gollum?'
'Hi! Gollum!' said Sam. 'Where are you going? Hunting? Well see here, old noser, you don't like our food, and I'd not be sorry for a change myself. Your new motto's always ready to help. ...
'Now, Gollum,' he said, 'I've another job for you. Go and fill these pans with water, and bring 'em back!'
'Gollum!' he called softly. 'Third time pays for all. I want some herbs.' Gollum's head peeped out of the fern, but his looks were neither helpful nor friendly.
'Wheew! Gollum!' Sam called and whistled softly. 'Come on! Still time to change your mind. There's some left, if you want to try stewed coney.' There was no answer.
Yes. The name "Gollum" goes back a long way, back to shortly after he found the Ring. See The Shadow of the Past:
They kicked him, and he bit their feet. He took to thieving, and going about muttering to himself, and gurgling in his throat. So they called him Gollum, and cursed him, and told him to go far away; and his grandmother, desiring peace, expelled him from the family and turned him out of her hole.
This establishes that not only is he aware of it, but that he also must have told Bilbo that "Gollum" was his name - since Bilbo was the only one he had spoken to (up to a certain time, of course), how else could others have known to use this name? How could Frodo have known during his discussion with Gandalf?
So - he wasn't only aware of it, he had accepted it as his given name by the time of The Hobbit.