I cant find any proof that battle axes were a thing, and from the history i know i cant find any record of such weapon. If they were a thing, who would have used them?
Yes. We've to archeologically evidence of them, for example the Royal armouries have a collection of norse battle axes brought up from the thames ad we have artistic evidence like depictions of them in the bayeux tapestry. They crop up with various designs in lots of places around the world, in early middle ages europe as i have already implied Scandinavian and Anglo-Saxons used particularly two handed great axes, typically they are particularly associated with the class or more professional warrior retainers in this context like Huscarls. Later on as the medieval period progressed you see them splitting into poleaxes and horseman's axes generally which keep being used throughout the period. For example famously Robert the Bruce carried one with him at Bannockburn. Meanwhile towards the east and more recently Tabar axes originating Iran were used in various parts of the middle east and India up until the early 19th. I believe this is also a higher status weapon based on the hollow metal handle design with longer handled infantry versions and shorter handled cavalryman's axes particularly commonly.
There are plenty of other examples where this kind of thing has been used scattered across the world, unsurprisingly given wood axes are common tools world wide with stone examples going back a long way, so adapting this kind of cutting tool to be purpose made as a weapon isn't a particularly large jump for lots of different people in lots of places to either made independently or inspired by other examples. One of their main values as a weapon is that compared to a sword they provide a fair bit of heft at the chopping end of the blade that gives it a good amount of cutting power against certain kinds of armour and of course flesh and breaking bone. More basic examples are also relatively easy to produce compared to a sword for people for whom good metal work was an expensive commodity although there were versions that developed over time with greater levels of complexity and more subtle elements in their design that would have been harder to create in return for various desired qualities.
There has been a very diverse variety of battle axes throughout history. Some examples:
From the Neolithic and the Bronze Age, we have axes like these:
https://www.britishmuseum.org/collection/object/H_2005-0501-80
https://www.britishmuseum.org/collection/object/H_2005-0501-74
which were probably made primarily for fighting. One difficulty with identifying objects as "battle axes" is distinguishing them from axes-as-tools and purely display/ceremonial/ritual axes. In this case, these axes are more hammer-like than cutting tools, so don't appear to be axes-as-tools. They might have been made as hammers-as-tools, but most old stone hammers were unhafted - it takes a great deal of work to make a hold for the haft in heads like these, and working hammers lead hard lives and break, wasting the work of hafting them. This suggests that these were made as weapons. It would be reasonable to call them "pointed stone maces" rather than axes, but "battle axe" is their usual name.
Battle axes with similar compact and stout heads continued to be made into recent times:
These are convenient sidearms, often carried by cavalry in addition to a sword, lance, and bow (or sometimes gun). The examples above with weight given average about 800g - despite their thick heads, they are small, and not heavy. The stoutness of their heads make them close-to-indestructible in battle, allowing their wielders to freely use them against armoured opponents (again, "pointy mace" is a reasonable alternative description).
Bronze axes from ancient Egypt and Western Asia show one often-important advantage of axes:
Axes with heads like these provide a long cutting edge and good reach for the amount of metal used, compared to swords. These axes can be thought of as cheap alternatives to swords. Again, we see axes designed for long cutting edges and reach with a minimum of iron/steel in more recent times:
For the more recent ones, reducing the cost by using less metal is often unimportant - the amount of metal is reduced to reduce the weight. Light weapons are fast weapons, and having a fast weapon that will let you hit first is important. Are lightweight axes less effective than heavy axes? They will generally be less damaging, but the weapon that hits the opponent is much more effective than the slow and clumsy heavy weapon that they avoid. Here is an example of a lightweight axe like some of the above example in use:
Comparing this axe with similar examples, I estimate the weight to be about 500g.
With some of these axes, the depth to which they can cut easily is limited by the haft, and how far forward the edge is from the haft. Once the haft hits, it is difficult to cut deeper. Another style of battle axe is designed for deep penetration, while remaining light. This is typically done by having a very short cutting edge. Axes like this appear in the Bronze Age:
and continue to recent times, notably in Africa:
https://www.britishmuseum.org/collection/object/E_Af1893-0804-31
https://www.britishmuseum.org/collection/object/E_Af1933-0208-1
To return to the earlier point about distinguishing between axes-as-tools and battle axes, some axes have features that are useful on weapons, but reduce their functionality as tools. This makes some axes easy to recognise as weapons. For example, these axes:
have spikes that are useful weapons but poor tools. This axe:
has a steel haft, which is more common on battle axes than tools. These axes:
https://www.britishmuseum.org/collection/object/A_As1878-1101-423
https://www.britishmuseum.org/collection/object/H_1838-0110-2 (a classic "Dane axe")
have heads that are thin and light for their size. These axes have long cutting edges, and long heads as measured from haft to edge (allowing deep cuts). To keep them light for agility, the heads are thin. Tool-axes of comparable size are usually much heavier.
who would have used them?
As noted above, small axes were useful sidearms for cavalry. They would also be useful sidearms for archers and spearmen (perhaps as a cheaper alternative to a sword). The larger axes, and especially two-handed axes, provide reach, often more than a sword, while being much more effective at cutting than a spear.
As swords became more common, battle axes appear to have become less common as infantry weapons, suggesting that they were often a sword-substitute. However, as swords became more common, a variety of cutting polearms such as bills and halberds became widely used on the battlefield. While these polearms are not usually considered to be axes, and are often much heavier than even large two-handed axes (e.g., 2.5-3kg vs 1.5-1.8kg), they often include an axe-like blade (e.g., halberds).