I am wondering if it's possible to determine if any SMS'es I get that are suspicious, can be checked if they really come from the sender without actually calling the number and asking the person behind that number?
Sure, if you set up some way to do this. An obvious way would be to use a "code word" known only to the sender and the recipient. If the message contains the code word, it's authentic. If not, it doesn't.
Suppose I have an application called "SMS Sender" on my laptop and it's connected to an SMS Gateway. I send an SMS from any number I like - to any number I like. How would the other person/party know if it's not the real sender but only the display number?
They wouldn't know whether it's the real sender or only the display number unless some way to accomplish this is set up.
Can I check this in any phone in some way or would I need to go ask my carier if the SMS is from the number that it told me it is from?
Your carrier doesn't know. The sender's carried does, but they probably won't tell you.
Or isn't my carier even able to determine if the sender is really the display number or a maniak pretending to be that number?
Your carrier might be able to tell under some circumstances, but generally only the sender's carrier will know for sure.
Like telephone, email, and physical mail, SMS just doesn't provide a reliable indication of origination. Again, like telephone, email, and physical mail, if you need such a thing, you need to arrange it with the sender. There are plenty of messaging applications to do provide reliable sender information.