I have never thought of time as being a real "thing". To me, in the physical world, there is no such thing as time. Particles, matter, everything, simply interact with each other and go on. Time is really just a psychological phenomenon that humans and perhaps other creatures perceive and use to keep track of events. Is my belief normal, or even scientifically accurate? Am I wrong, is there something I'm missing here?

Perhaps it'd help to think of time as a number rather than an object. The same way we consider distance. If you and I stand three metres apart, that distance of three metres doesn't exist as a physical object. You can't pick up three metres and put it in your pocket.
As humans we're more used to the concept of distance because we can control it, and it is elastic. If I take four steps towards you, I am now one metre away as opposed to three metres. Time is elastic too; we just don't ever move fast enough to notice relativistic effects so we don't view time in the same way as distance. We should do, though.
The scientific principle we should be discussing is entropy, otherwise known as the second law of thermodynamics. Barring a few other rare forces, it is the only physical, universal law that REQUIRES temporal direction. It is the reason that eggs break but never spontaneously reassemble, and the reason that it is impossible (yes, impossible) for all the air particles in the room to suddenly compress into a corner of the room. Entropy is the arrow of time.
You made me think about it again, though. My brain is too tired and hungover to put too much effort into the following question but I'm interested in your answer.
Can you measure something which doesn't exist?
Not a trick or a trap, by the way.
I think the question that the OP is positing refers to the arbitrariness of time, or rather our structuralization of time, which is what you were getting at. As far as your question is concerned... my friend once asked me if I believe love exists. I said it must, if only for the fact that someone felt it strongly enough to name it that.
Eschewing any talk of quantum mechanics and collapsing wave-functions (I'm assuming you're not trying to go there), no, I don't think you can measure something that doesn't exist. It's a bit of an odd question though, because I'm no longer sure what you mean by "exist". It's like asking if we can create a word that doesn't have a definition... or what the sound of one hand clapping is.
I take your point about love, too. Although I'd amend it and say that it has a name because it's important enough for people to 'want' it to exist, not necessarily because it does. Same goes for God, ghosts, energy from crystals
I debated long and hard with my metaphysics teacher over this topic. He always played the Devil's Advocate. Probably the best prof I ever had, he had a gift for starting good discussions in class, using often ridiculous examples.
Fucking Dr. Pepper.
Here's something didactic. Time was quantified in the 1400s in order to keep track of business. So yes, the conception of time was a means to regulate events.
As for particles. Well, what does it matter? haha.
In all seriousness, time is a phenomenon that we humans want to grasp. Time is a powerful thing. Laws are built around it, think thermodynamics. Everything decays due to time, however, things evolve due to it as well.
The way you perceive time is rather interesting in its detachment. I'm not at liberty to say how correct you are or how wrong, frankly, you could be right. All I have to say is that I like your opinion.
Our circadian rhythms calculate time without any external input (the sun). It's not psychological. All living things on the planet 'calculate' time.
It looked like i had bernards watch, everyone in the room just froze. This lasted a few minutes but it felt like forever.
Fucking weird. I could still move about... Time is crazy.
When Einstein came along he showed using his equations that time was NOT constant. That people in different situations could theoretically feel different passages of time. Especially when dealing with the speed of light. As you approach the speed of light theoretically the equations tell you that to keep everything balanced time has to slow down. This in my view is what makes Einstein so great. Introducing a completely new way of modeling gravity time and motion that went against the 'establishment'.
Let's go back to our example, I am on the planet earth and you are on a spaceship approaching the speed of light. Time for you is now slower than time for me. While an hour passes for you, many hours pass for me. When you finally return to me you will notice that I have aged more than you. This shows that time is not just an arbitrary measurement that humans use to mark the passage of time, but a changing entity, a changing quantity that is defined by different variables and conditions.
So basically if he was right, neutrinos would be travelling back in time lol.