I'm hoping you are not as dense as this post made you out to be.
Is the KV-1 able to Kill all Axis mediums tanks, including the premium mediums like the PzIV J?
yes KV-1 has chance to win even if it is flanked.
Then your sentence:
"Is it AT medium+ capable? No"
is simply wrong.
Even if you want to talk about being supported the KV-1 is simply superior to the PzIV since it can take allot more punishment while its supporting units can dish out the damage.
Ok you are not as dense. However, my point is not that it's not sturdy or whatever. That all stands. KV1 is by all accounts superior to both P4s. And it's not AT+ capable. AT+ capable units are T34-85, ZiS, Pak, Raketen, JP4, Stug, Panther, Jackson, E8/76, M4, etc. Those units are AT+ capable. It does NOT mean that KV1 can't go against any tank; heck, my elefant was once taken down by 3 KV1s in a really fun 1.30h game. You can use KV1 to take down P4s, but you usually won't since it's not as easy and you might run into a wild Pak or raketen or a wild untamed faust. If I'm understanding it correctly, you say that it's AT capable because it can go toe to toe with other tanks thanks to it's sturdiness... which is correct, but not my point.
My point is the following:
You are going against 5 targets. Let's say that they linearly scale with survivability and inversely proportional to their damage output.
So unit 1 has 1000 hp and 100 dmg
Unit 2 800 hp and 200 dmg
3 600/300
4 400/400
5 200/500
You are going against those units. Now in an average MMORPG or RPG or whatever, the unit one would have some sort of AGGRO skill that would FORCE you to engage it for a fixed amount of time. It's a tank, it can take punishment but also survive it. Unit 5 goes down easily but also takes down easily.
KV1 is the unit 1 in such conditions, but without the Aggro skill. It's damage output vs tanks is inferior to other Axis tanks, medium or otherwise. So, as an axis player, if you can you will not try to focus KV1 down first, because you know it won't massacre your vehicles like SU85 or something like that will. What you will do, I hope, is start taking down units that are easy to take down, one by one (I think it's self evident why taking down units one by one is better than letting everyone shoot at everything). So KV1 will have supporting infantry and your P4 will have supporting infantry and KV1 will probably fire at your supporting infantry but you won't make the mistake of trying to fire your P4 at KV1, instead if you have a pak or shrecks, you will use them, but only if KV1 is solo, if KV1 also has T34 close by, you focus T34 first.
So, while KV1 is a great tank, superior to stock P4s due to the fact that it's tanky AF, it doesn't have enough firepower vs tanks to make it a menace.
And if your shrecks bounce, well it's just tough luck and the RNG gods were not on your side (does not mean it's armour is too high), because the penetration chance is high enough vs it for the role it's playing. Same as how elefant and jagdtiger are expensive AF but also hard to take down with 400 frontal armour. You lose some, you win some.