Answer time! For the first part anyway.
Spoiler:
Our hero can, of course, escape the desert. He just needs to arrange to get to a point two days walk away from the oasis with four days' water.
Since he can only carry four days' water at a time, that means he'll have to have put some water at the two day point to pick up when he gets there to replenish his stores.
Now, he can't just carry water to the two day point directly from the oasis, as when he gets to the two day point, he'll have used up two days' water, and will only have two days' water left, which he'll need to get back to the oasis.
So he must make a cache of water between the oasis and the two day point. The obvious first place to try is the one day point. He can start off with four days' water, walk for a day, leave two days' water there, and walk back to the oasis, using up his last day's water.
Now - how much water will he need? Well, he needs to carry sufficient water from the oasis to supply all the days he's in the desert. That's four days getting out after he's set up the caches of water he needs. At least two days setting up the intermediate cache and at least three days setting up the two mile cache. That's an absolute minimum of nine days of water. But if he only goes back to the oasis once, he can only obtain eight days of water.
So we can see that he must re-visit the oasis at least twice, getting him a total of twelve days of water.
Since we've found he must re-visit twice, let's get all that done first. He goes back and forth between the oasis and the one day mark, carrying out water each time. At the end of the fifth day, he's at the one day mark with seven days water. He can now go out to the two day mark, leave two days water behind, and return to the one day mark at the end of the seventh day. At the one day mark, he now has only three days water. On the eighth day, he takes this water, collects the two days' water that he stored there, and now has four days' water with him, allowing him to reach the edge of the desert at the end of the twelfth day. Setting out the timings, it looks like this. I don't mention it in the table, but every day a day's water is used up.
Day 1: Take four days' water from the oasis. Walk out for one day
Day 2: Leave two day's water. Walk back to the oasis.
Day 3: Take four days' water from the oasis. Walk out for one day
Day 4: Leave two day's water. (Now four here in total.) Walk back to the oasis.
Day 5: Take four days' water from the oasis. Walk out for one day. (Now seven here in total.)
Day 6: Take four days' water from the seven you have, and walk out for one day.
Day 7: Leave two day's' water. Walk back to the one day mark.
Day 8: Take the three days' water you have and walk out one day.
Day 9-12: Take the two days' water you brought with you, and the two days' water you left here on day 7, and walk out of the desert, escaping the desert at the end of the twelfth day.
Unfortunately, I ‘forgot’ to mention that there's a literal deadline. The oasis is due to be the site of a nuclear test in 285 hours. (12 days less three hours.) There's an observation bunker at the edge of the desert where he'll be safe, but he must get there before the bomb goes off, or he will be killed by the blast*. The solution above doesn't get him there in time. Can you get him there in time?
Big Clue:
*Yes, it's an unfeasibly large nuclear test