It'd be nice if the question actually said, but I'll assume you're struggling with roundhouse kick.
But somehow I'm feeling like it's difficult to raise my leg.I'm also not very good at setting the distance for a kick, and I'm either extending it too late or too soon.
I think you should take a good look at your stance. You want the stance to be reasonably square on (so you can comfortably rotate your back hip past the front hip when throwing a cross/reverse punch, and rotate the hips back for a jab, without losing balance). The back foot should be facing about 30 degrees clockwise from a straight line to the target (NOT sideways, which tends to leave the hips too side-on to the target). You want your weight settled downwards with knees bent. When you raise your kicking leg, the knee should be able to arc comfortably towards the target as the hips rotate. Think about that knee arc into the target, not just lifting the leg for its own sake: you should not be lifting the knee vertically behind you then trying to rotate the hips. It's perfectly fine (even encouraged in many kickboxing / MT schools when kicking for power), to step the front foot slightly forwards and out (say 20cm, 45 degrees counterclockwise from a line to the target) before starting the kick, which creates some momentum to drag the hips and kick through (this does telegraph your kick though).
With all those things in mind, turn to Sean Duggan's advice and find a heavy bag. Make sure you're making solid contact through the bag.
More generally, in martial arts/sports it's an oft-useful practice to consider how you want a technique to land, and work backwards from there to understand how the stance should be to comfortably launch you into the correct motion. You can do this quite literally (start with the striking limb out and return to the fighting stance). Keep the movement minimal, strong and balanced.
Don't get fixated with this training partner. The problem is obviously with your technique, not him. If anything, it's probably good that he's forcing you to work on the body mechanics for the lower kicks, because all this "I can easily do the splits thing" suggests you might be more inclined to focus on high kicks and ignore the lower ones. Given your flexibility is already good, you should work on your strength and stamina - try to get comfortable throwing say 20 kicks into the bag in rapid succession (then do the other leg, rinse and repeat).
Sean's also right that your instructor's "you are not good enough" is a huge red flag.
I also need to improve my balance on my front leg. I am an orthodox, so I keep my left leg forward. The idea is that when I get a low kick, and I don't check it, I need my leg to be stable, and not to flinch.Should I put my weight on it?
Most martial arts similar to kickboxing recommend somewhere between a 40/60 and 20/80 split between front and back leg, so yes there should be weight on the front leg, but not so much you can't lift it easily to check or kick. You should make sure it's comfortable to quickly thrust into the ground to drive your body backwards too. (In MT in particular, you will sometimes see fighters put less weight on their front foot - bouncing that leg up and down as they consider throwing a teep or front leg roundhouse to an opponent on the edge of kicking range, but that's movement's not used once the opponent's closer).