Yes, both those pushes are valid.
When you push a creature, each square you move it must place it farther away from you. [RC 211)
Now, 4e measurements use taxicab geometry, or Chebyshev distance, so the effect is... often nonintuitive. I'm pretty sure 4e floors are constantly shifting hyperplanes. My players have developed the mantra, "Circles are squares, squares are circles, and never draw a triangle on the grid!"
How to tell if your push is valid:
To tell if a creature is moving closer, further, or the same distance, count the number of squares the creature must enter to be adjacent to you if he starts in the first square, or the second. The more squares he must traverse to reach you, the further he is from you.
You also need to have line of effect to every square you push him into. He can't enter blocking terrain, and ignores difficult terrain.
The flavor text describing a power's visual appearance has no effect on the power's mechanical application.
Weird.
This actually means that the squares you can legally push a creature into differ, depending on if the creature is head-on with your square or adjacent by corners, because any square adjacent to both you and him is invalid for a push when he is also adjacent to you.
However, so long as you fulfill the above prerequisites, you're free to bounce the guy around zig-zag-style; nothing says the movement needs to describe a straight line (and given 4e geometry, I'm not sure what a straight line would look like anyway).
For completion's sake, pulls and slides:
For pulls, each square must be closer to you than the last, and slides don't care what your position is related to the target.
It's out of place in 4e, but 5e is a different game and it's not out of place in any way. You're not missing a rule, just importing expectations into the game that aren't part of it.
To avoid being pushed off cliffs or onto spikes in 5e, you do what people do: stay far away from cliff edges, pits full of impaling spikes, and other lethal hazards. Adventurers will often choose to forego common peasant wisdom, but that is the price of adventure and a chance at fabulous treasure. Nobody said adventuring was a safe occupation.
Best Answer
From the DMG/Compendium, the important part for your question:
And from the DnD FAQ ( http://wizards.custhelp.com/cgi-bin/wizards.cfg/php/enduser/std_adp.php?p_faqid=1396 ):
The resultant answer: For zones, base DnD rules do not give a saving throw to avoid being put into the fire. For pools of acid, you are given a saving throw to catch yourself before it and fall prone.