You do have some errors in your little rant. While firearms are easier to acquire in the US as a rule, there are still some restrictions and forms you need to fill out. Plus there can be a near byzantine set of laws that each state and even cites can pass to further restrictions on purchase and ownership. It might be the biggest issue about firearms ownership that there are few national laws for enforcement. It's mostly up to each state and city regulations and enforcement.

Convicted felons are pretty much barred from firearms ownership across the US. The only real exceptions might be a billionaire who can buy anything. Or perhaps it's just easier to pay some else to shoot people for them.

Everyone has to fill out a Form 4473 which is a universal federal background check against a data base to see if you can legally own a firearm. It is an electronic background check done at the time of sale and transfer. It can take a few minutes or a few hours to get done. And you can be disqualified for a simple misspelling or even if your name is similar to a some who is barred from purchase. Then it's up to you to get your name cleared. All and any firearms purchases through a dealer MUST have a Form 4472 attached. And the dealer must keep a record after the sale for a fairly long period of time. A good number of states have further restrictions and requirements on the purchase and ownership of firearms. Which require further state background checks and issuance of a special card to buy a firearm. And individual cities can impose further restrictions yet.

Minors, under 18 years of age, (a few are 21), in the majority of states cannot legally buy a firearm. And are generally only allowed to handle or use a firearm with an adult present - some exceptions would be during a hunting season and only when hunting. But even then, there will be an adult somewhere around.

Carrying a handgun publicly, with the exception of a very small number of states, is very controlled. Some states, like California or New York are quite restrictive to the point that pretty much only wealthy people can actually afford to pay for all the hoops you might need to jump through to get such a permit. A tiny number of states, like Texas allow for common carry laws without a permit, (often called Constitutional Carry). But the majority of states require that you have taken a special class and then go through more special background checks by local law enforcement to get the permit issued. And these permits require renewals every few years with more background checks every time.

Again, I think the biggest issue is the lack of a uniform national set of laws and requirements for firearms purchase and ownership is what confuses everyone. States are considered to have most of the power to make many laws that the federal government can't over ride. Sometimes this is a good thing and sometimes not so good perhaps. But it's the system we have for better or worse.

you can buy this here in austria without a Waffenpass or Waffenbesitzkarte, you literally just need to be 18

https://www.brownells.at/PUMP-RIFLE-308-WINCHESTER-16-Pump-Rifle-308-16-TROY-INDUSTRIES-INC-Black-Pump-Action-101-Round-33-36-85-lbs-16-None-Removable-Polymer-1-x-10-Round-Medieval-Muzzle-Brake-100041732

EDIT: yeah, I thought I had the wrong numbers in mind

EDIT1: also, the Waffenpass (the thing you need to carry them in public isn't hard to get https://www.oesterreich.gv.at/themen/gesetze_und_recht/waffenrecht/2/Seite.2450900.html#Voraussetzungen)

In the US meanwhile [...] [a]nd that without any background checks, psychological reports, justifications, approval required, without anything like that. In many states even convicted criminals can get guns like that.

If you're talking about buying a firearm from a store, that's simply not factually correct.

Every single firearm sold by an FFL holder must have a form 4473 filled out, and each person buying a firearm must go through a criminal background check. ANY felony conviction that could have sent you to jail for more than a year--regardless of whether or not you got jail time--permanently bans you from owning a firearm until the conviction is expunged (and in many states, your gun rights need to be proactively reinstated). Any misdemeanor domestic violence conviction will likewise bar you from legally owning a firearm, as will having an active retraining order. Being involuntarily committed to a mental facility will bar you from ever owning a firearm at a federal level (without a judicial proceeding to restore your rights), and being voluntarily admitted will cost you your rights in some states.

Keep in mind that these are federal regulations that supersede any state or local regulations. A state can not opt out of the NICS or decide that gun stores don't need to comply with BATF regulations. The only "exception" per se is that, in my state, a carry permit means that the gun store doesn't have to send in form 4473 for approval; you've already passed a more stringent background check--including fingerprinting--so it would be moot. You do still need to fill out a form 4473, and the gun store is still required to retain a copy, but the instant background check is deemed irrelevant.

I'm not arguing one way or another but I want to clear up some very common misconceptions about US gun laws.

in some states, you can get semi-automatic weapons (which are completely banned in Austria) in a shop in just minutes. And that without any background checks, psychological reports, justifications, approval required, without anything like that

This is just blatantly untrue and I wish people would stop parroting it. If you go to any shop you need to pass a federal background check to buy any non-vintage firearm (pre-1899..not exactly a ton of those floating around). The exception here is private firearm sales, i.e. I go to Craigslist and sell a rifle or handgun. The law states the seller has to have no reasonable cause to believe they would be an unlawful possessor (weak, yes). With that said, almost half of the states (22 per Wikipedia) have implemented state-level laws requiring a background check for private sales.

In many states even convicted criminals can get guns like that.

Again, objectively untrue. You are not buying a firearm from any legal, licensed dealer in the US without going through a background check. And a violent criminal offense will get you barred from purchasing. For the 28 states without laws around private sales, the seller can be federally legally liable if they sell to someone that is not legally allowed to have a gun and they use it to commit crimes.

In the US, guns are sometimes a presents for kids which they can just…own and use (while in Austria everything is obviously 18+).

No, a child cannot legally own a firearm. The parent can purchase and own a firearm that they are allowed to use, but they do not own it. In many states if the child hurts themselves or others with such a firearm the parents will be held liable, many states have laws around safely storing firearms when children are around.

While in the US (in many states), you can just carry any gun around in public whatsoever. So even if the police sees you having weapons in public, they can’t / don’t do anything about that

In most states if you don't have a license to conceal carry and you do you are breaking the law and can be charged. I'll say this one isn't entirely false but heavily depends on your state.

A large part of why this issue gets nowhere is that neither side can even agree on what is true today, rather than what should be true to bring down the issue of violent crime. If one side says "They're totally unregulated you can just buy one off Amazon and start blasting. We have to do something!" The other side is gonna think "Well they obviously have no idea what they're talking about, no point in listening to what they have to say"