There are several types of corporations in the US, and you can see them explained here:
All of them have in common that they limit the shareholders’ personal liability in legal or business disputes. The only differences are details of taxation, etc.
“Inc.” means “incorporated”. All it says about the company is that it has the legal status of a corporation (i.e., an artificial person), and that its owners’ personal assets are protected. In the US you will also see “PC” behind the names of many doctors. This means “private corporation” and that the doctor’s personal assets are protected against liability.
“Ltd.” has no legal status in the US, as far as I know, and it’s just added to the names of corporations to make them sound “classy” and maybe even “British”. You will also find that some corporations use the word “Unlimited” at the end of their business name, such as “Movies Unlimited” or “Mustangs Unlimited”. This has no legal significance and is just a pun on the British term “Ltd.” It’s supposed to indicate that they provide an unlimited choice of products and services.
I don’t think Americans distinguish, in terms of trust, between companies that are regular corporations or LLCs.
Incorporation laws vary from state to state, so it’s not possible to say who can and can’t incorporate “in the US”. However, it must be possible for foreigners to incorporate here, because so many companies do it. Some foreign companies – including some German ones – have, in the past, basically turned themselves into de facto US corporations by expanding their American subsidiaries and reducing their European operations so much that their original “home offices” were really just branches of the US operation. Companies often cherry-pick the state they incorporate in, based on the advantages state laws offer to their type of business. Financial companies often incorporate in Delaware or one of a few other states that have no usury laws. Some companies incorporate in states that have no income taxes.
One big difference you’ll notice is that in the US you don’t need to have a large cash reserve to form a corporation. My understanding is that in Europe you need a lot of cash to incorporate, possibly as much as 50,000 euros. In the US you simply need enough money for the fee to process the incorporation papers, which varies depending on the state, but can be as little as $35. Another difference is that in the US you are allowed to explain the purpose of the corporation very vaguely, so that you don’t have to keep going back and amending the papers every time you enter a new line of business, the way people have to do in Europe.