This is the Kilenya series by Andrea Pearson.
As per the Goodreads synopsis for the first book, the protagonist, a teenager named Jacob Clark, is transported from Earth to the world of Eklaron.
We all have a choice—but we can’t choose the consequences.
Jacob Clark is chased down a path that takes him to another world—a world where he is a wanted young man. The Lorkon want to control him and the special powers he possesses. The people of the new world want him to save them from the destruction of the Lorkon. All Jacob wants is to go home, but even that choice has consequences. If he waits too long to decide, the new world and the one he came from will be destroyed.
As Jacob looks for people he can trust, he finds himself in the center of a fight for freedom—both for himself and the people he’s come to know. And when he meets a pretty girl named Aloren, everything he understands about his role on Eklaron is challenged.
Jacob has no idea where the path he chooses will take him, but once the choice has been made—bring on the consequences.
The plot of the first book revolves around the Key of Kilenya, one of two magical keys that enable one to go through any door, anywhere, regardless of their current location. The other key is later named as the Key of Ayunli.
He leaned forward. “I’m going to get right to the point. Two weeks ago, an object was stolen from our village. It’s a magical key that was made hundreds of years ago, along with one other, which has also been lost. This key was created to save a princess from an evil king.”
“What does it do? Open a treasure box or something?” Jacob asked.
“Yes—every treasure box ever made, and more,” Ebony said. “It’s a powerful instrument. When placed into any lock and turned to the right, it opens the door—or box—regardless of the spells or locking bolts used. When turned to the left, one can go through any door, anywhere, regardless of one’s current location.”
The Key of Kilenya (2011)
Jacob discovers that he has the power to sense other people's emotions, with each emotion being denoted by a different colour.
Most everyone figured out he could sense other people’s emotions, and now girls—why was it always girls?—asked him for help with their relationships. “Is he still mad at me? Does he like me? What’s he thinking now?”
The worst was when a girl approached him and asked what she was feeling toward him. He blushed and walked away. It had been many different colors—the ones that equaled infatuation.
It didn’t matter that he’d stopped asking other students how they felt. Word spread, and he was constantly stopped by new people as he went to and from classes.
The one time when this ability helped Jacob was on the court during lunch. He could always tell when people were distracted by strong emotions—fear, anger, love. He didn’t feel bad for zipping past players who weren’t really into the game.
The Ember Gods (2011)
He later discovers another power, referred to as 'Time-Seeing,' which enables him to view events in different time periods. He's told that with practice, he'll even be capable of time-travel.
“With practice, you’ll learn to master this ability, and then you’ll be able to take your Time-Seeing to the next level: Gussar. I suspect you’ll refer to it as Time-Travel.”
“Wait—did you just say time travel?”
“Of course. I’ll teach you how. Don’t try to figure it out on your own—there are many, many rules you must follow that will keep you from killing yourself.”
August Fortress (2012)
Moreover, he discovers that his 'parents' on Earth aren't his biological parents, and that his biological father was King Dmitri of Eklaron.
Jacob took the phone from his brother and dialed his dad’s number. He was directed straight to voicemail. “Dad, you and Mom need to come home right now. Was . . . was I adopted? I really need to talk to you guys.” He hung up and noticed the expression on Matt’s face.
“Adopted?” Matt asked. “You?”
“I’m sure of it—the journal is talking about a baby boy who’d been—” Everything hit Jacob at once. If he was the son of Arien and Dmitri, that meant he was royalty. He sank to the couch. A prince? Him? Then he remembered what Gallus had said when he and Akeno entered the shop for the first time. He’d called Jacob “Your Highness.”
The Ember Gods (2011)
The Key of Kilenya (2011) was later rewritten and republished as Forsaken Prince (2017), which Goodreads lists as the first book in a separate series, the Kilenya Chronicles. However, books 2-6 are the same as books 2-6 in the Kilenya series, so it appears that these two series are, for the most part, one and the same.
Since The Key of Kilenya and Forsaken Prince were both released within or before the period in which you said you read the books you're looking for, I'm unsure which of these two books you might've read.