No. Dumbledore didn't give Snape the full picture.
Dumbledore is a known liar who operates on a secretive basis.
"I knew my brother, Potter. He learned secrecy at our mother's knee. Secrets and lies, that's how we grew up, and Albus...he was a natural."
(Deathly Hallows, Chapter 28, The Missing Mirror).
It would be highly uncharacteristic for Dumbledore to entrust the whole truth to anyone. He preferred to keep both Harry and Snape partially in the dark regarding what he suspected about Harry and Voldemort.
"I prefer not to put all of my secrets in one basket, particularly not a basket that spends so much time dangling on the arm of Lord Voldemort."
(Deathly Hallows, Chapter 33, The Prince's Tale).
Dumbledore only gave Snape the information that Harry would have to die as a concession to Snape, who was complaining about being misinformed.
"I spend time with Harry because I have things to discuss with him, information I must give him before it is too late."
"Information," repeated Snape. "You trust him...you do not trust me."
[...]
Snape looked angry, mutinous. Dumbledore sighed.
"Come to my office tonight, Severus, at eleven, and you shall not complain that I have no confidence in you..."
(Deathly Hallows, Chapter 33, The Prince's Tale).
As a result Dumbledore gave Snape new information, information that he didn't give to anyone else. Snape alone knew that Harry would have to 'die' for Voldemort to be defeated. He did not tell Snape about the fact that Harry had a chance of surviving Voldemort's attack because Harry's blood flowed in Voldemort's veins. He told Snape what Harry needed to know. Nothing more, nothing less.
Additionally, it's unlikely that Snape had deliberately meddled with his memories to mislead Harry. We see Slughorn attempt to do this and the results are utterly transparent.
"As you might have noticed," said Dumbledore, repeating himself behind his desk, "that memory has been tampered with."
"Tampered with?" repeated Harry, sitting back down too.
"Certainly," said Dumbledore, repeating himself behind his desk, "that memory has been tampered with...He has tried to rework the memory to show himself in a better light, obliterating those parts which he does not wish me to see. It is, as you will have noticed, very crudely done..."
(Half-Blood Prince, Chapter 17, A Sluggish Memory).
If Snape was trying to hide information from Dumbledore I think we would've seen something similar to what we saw with Slughorn's memory - lots of fog and a blank space where the real memory lay hidden. We don't see this effect. I think Snape was being honest with Harry.
Another possibility, of course, is that Dumbledore told him the truth on a later occasion and Snape just didn't include that memory (in which case Snape would obviously be the one withholding the information, possibly at Dumbledore's request).
This is, naturally, impossible to disprove. However, we have no record of such a conversation in canon and the reasons I've given above make it unlikely.
Why wasn't Harry told about his chance of survival?
This is much more difficult to answer. Dumbledore only hints at the motive for his secrecy (beyond it simply being second nature).
The question moots the possibility that Dumbledore was worried that Voldemort might read Harry's mind with Legilimency. This was absolutely a concern of Dumbledore's.
"Good. Very good!" cried the portrait of Dumbledore behind the Headmaster's chair. "Now, Severus, the sword! Do not forget that it must be taken under conditions of need and valour - and he must not know that you give it! If Voldemort should read Harry's mind and see you acting for him -"
(Deathly Hallows, Chapter 33, The Prince's Tale).
Even if this were the only reason then it is a sound one. If Voldemort performed Legilimency on Harry and saw that he was acting according to Dumbledore's instructions then he would certainly have exercised more restraint and caution in the Forbidden Forest.
Would Harry have survived if he'd known that his death might not be permanent? Yes. It didn't matter what Harry knew. The key to his survival lay in the fact that Voldemort had his blood in his veins, tethering him to life. Yet the fact that Harry died voluntarily seems to have been important somehow.
"But you're dead," said Harry.
"Oh, yes," said Dumbledore matter-of-factly.
"Then...I'm dead too?
"Ah," said Dumbledore, smiling still more broadly. "That is the question, isn't it? On the whole, dear boy, I think not."
They looked at each other, the old man still beaming.
"Not?" repeated Harry.
"Not," said Dumbledore.
"But..." Harry raised his hand instinctively towards the lightning scar. It did not seem to be there. "But I should have died - I didn't defend myself! I meant to let him kill me!"
"And that," said Dumbledore, "will, I think, have made all the difference."
(Deathly Hallows, Chapter 35, King's Cross).
Dumbledore states but he doesn't explain. He later says that Harry has a choice as to whether to go back or go on to some sort of afterlife. He doesn't mention the sacrificial factor in Harry's death as being an important reason why he has this choice. It seems that Harry's blood and Lily's sacrifice living in Voldemort would be enough to ensure Harry's survival whether Harry died sacrificially or in the middle of a fight. Dumbledore doesn't state why Harry sacrificing himself might be important, just as he never adequately explains why Voldemort had to be the one to kill him. Of course, Harry doesn't need to fight since any attempt to kill Voldemort is futile whilst the last piece of soul survives within him. Perhaps it's simplest to avoid Harry getting into a fight he couldn't win, and the easiest way to do that was to make him think that there was no hope. Yet I suspect that Harry would've liked to have known that all was not lost. All that was gained by Harry dying sacrificially, as far as I can tell, was that his 'death' protected the forces of Hogwarts from the spells of Voldemort and his forces. This is a strategic coup in the battle but it doesn't have anything to do with Harry's survival. Nevertheless, Dumbledore states, for reasons unknown, that Harry dying sacrificially "made all the difference".
In conclusion, Dumbledore kept back the full truth from Snape but the impact this may have had on Harry's survival is less clear.
