Can a player character throw a hand-axe upwards over a wall, into an enemy group that is on the opposite side of the wall? If so, what penalties would apply?
Putting aside the rules... I don't see why not.
But, if you want to do it effectively, there's a few more issues.
Firstly, assuming the wall is opaque, you're effectively throwing it blind. Determining the effectiveness of this would be pure chance, and I'd say it'd be entirely up to the DM on how that chance is handled. You may be able to roll for how effectively you threw the axe, but even a nat 20 roll wouldn't determine a hit.
The DM would likely factor in the direction you're throwing it, ask what you're using as a frame of reference (like sounds, or flickering shadows) and likely ask for a perception check to identify your target, then depending on what the target is doing (standing still? moving about?) determine the accuracy of your perceptive abilities.
While this might be a bit of overkill, but the skill used to determine damage is used to determine the damage of a direct hit. For example, you use Strength to bury an axe in your opponent's skull. In this instance, that would be completely irrelevant. It would be entirely up to the weight of the weapon to deal the damage.
So, in this instance, let's say the wall is 15ft high. To be safe, you throw it roughly 20ft, just to be sure. On the way back down, you'd rely on the weight and speed of the weapon to deal the damage. At ~20ft off the ground, it'd only reach ~60ft/s/s, which means that at ~2lb, the damage dealt would be a little more than a nasty knock, and that'd even be if it hit the right way around!
So all in all, this would likely be either completely useless, or used as a distraction more than anything. The effort involved in even getting this to work properly is over the top, and it would therefore make sense that anything in Full Cover (as Nyoze has explained) is impossible to hit. At this point, it'd be safe to say that physics and D&D do not mix well.
Thanks to Nyoze for helping me figure out the math on this one
You can throw an axe over a wall, but it isn't an attack.
The wall provides total cover and "A target with total cover can't be targeted directly by an attack..." This means that tossing a weapon over the wall can't be resolved as the Attack action.
However, if you want to attempt something that isn't covered by the rules, you are explicitly allowed to Improvise an Action, per the combat chapter of the PH. How the success of this action is determined and the result of success are at the completion discretion of the DM. They could call for an ability check from you, make a saving throw for the target, or even just say that it bounces harmlessly off the helmet of the target. If the action is successful, the DM could determine that the target is dealt 1 damage, 1d12 damage, or even 1d6 for every 10 ft of height of the toss damage. It's completely up to them.
Rules as Written - No
As per the Cover rules, any attack at a creature that you aren't able to see, due to an obstacle completely obscuring the enemy, is treated as if they had Full Cover.
A target with total cover can't be targeted directly by an attack or a spell, although some spells can reach such a target by including it in an area of effect. A target has total cover if it is completely concealed by an obstacle.
In the case of your example, you won't be able to target or hit any enemy with the thrown axe.
Think of this as a real world example of throwing a ball over a fence at a friend who is trying not to be hit. Any momentum in the throw will be lost on the way up, and you wouldn't know where behind the fence your friend was standing.
Yes, one can toss an axe over a wall. But it is not an attack.
There is no accuracy, precision or strength in such an action. You won't be rolling any d20, but there is a chance the axe would hit someone on the other side. There are 3 sides of the action: making the axe go over the wall, finding where it will fall and evading/dealing damage.
This is how I'd adjucate the action:
Make a Strength(handaxe) check DC10 to make it go over the wall. This is not an attack roll, just a ability check. Maybe the character can't even lift.
Select a random space on the other side of the wall. The DM should see where it might land, choose some squares, assign numbers to them and roll a pure dice.
If the space is occupied, allow the creature on the other side to make a Dexterity (DC 11) save. Now the falling axe is a "setback trap". Let the damage be the normal handaxe damage for the characer. On a successful save the character evades the axe and it hits the ground.
Short answer: yes.
It breaks down into two parts, since it can't be treated as a normal attack.
On the player's side of the wall:
The player is trying to throw an object over a wall. It's not an attack on a specific enemy, so it is a feat. You can do the checks you think are most appropriate to test if they succeed. Keep in mind that just throwing it anywhere over the wall is easier than throwing it at a particular square on the other side of the wall. If they try to aim at a particular square, then the checks should be more difficult.
On the far side of the wall:
An axe falls from the sky. Does it hurt anyone? If the player picked a square and passed the check, then it lands there. If not, then you can use some die rolls to randomize where it lands. From there you might want to treat the falling axe as a dangerous trap with the character needing a passive perception check to see if they notice the axe before it hits them. Even if they notice they probably then need to make a dex check to evade harm. If it hits them then you as DM have free reign to calculate how much damage a falling axe does. The player's stats shouldn't be used here, because the player didn't make a standard attack. What happened on the other side of the wall is irrelevant now. All we care about now is that an axe weighing X pounds fell from Y height and landed on a guy.
I would allow it. However I would also apply modifiers such that the player throwing the axe would get a blindness modifier and any strength modifiers are removed.
When you throw an object it is possible to aim if you have a grasp of how and where it will impact a surface. Throwing with no vision impacts the aim. In dnde5 it could also mean that there is full cover, but that would not be the case if one used something like a siege weapon (think mortar), so I discount that as a valid conclusion.
When you throw an object over 45 degrees, the launching speed (aka. the strength modifier) is affected and gravity comes to account for most of the impact.
Thus if you are able to (by other means than vision) aim at a target, you would be able to hit it. It is important to note that a weapon such as a hand-axe is difficult to throw as it tumbles in the air and the location of impact is will have to be "blindly" aimed at around four foot from the ground at a certain location. There is the option to add the 50% chance to miss due to blind fire, which should apply in the case where the wall is not transparent.