This is more of a pure physics question and I'm not sure it would do well there either, but I can touch on some basics.
I am at gym - a regular one, on Earth. Escape velocity of Earth is
around 11.2 km/s. Yet I can lift weights up say one meter even at
slower speed, say one meter per second.
You can lift a weight at the gym and hold it there, but only by doing constant work. If you let go, it'll fall to the floor and make a noise, and perhaps get your membership revoked.
You could, in a thought experiment anyway, climb a ladder up into Earth's orbit, past the satellites and keep going out past Earth's Hill Sphere, where the gravity of the Sun would take over and you could let go and part ways with the Earth. It makes no difference how fast you climb the ladder but that would require applying continued work. Applying work has nothing to do with escape velocity. Escape velocity is like throwing a ball up into the air, once you let go of the ball, it has an initial velocity, and the "what goes up must come down" law takes over.
What comes up must come down, assuming we ignore air resistance, is a good way to explain escape velocity. If you throw something faster than 11.2 km/s, it won't ever come down, it'll keep going up until it escapes the Earth.
Escape velocity doesn't apply to rockets which apply several minutes of thrust to escape Earth's gravity and enter orbit. (technically in orbit is still in Earth's gravity well, but it's escaping a significant part of Earth's gravity it in order to remain in orbit) - see xkcd
Escape velocity only applies to initial velocity and it ignores air resistance.
Next, consider me hovering one meter above event horizon. As far as I
know, for non-rotating BH this is totally possible. Similar to earth,
I could lift weight up one meter at speed less than escape velocity
(which at this point is speed of light minus x).
You can lift a weight 1 meter in your gym because the Earth's surface is holding you up. An event horizon has no surface, so there's nothing for you to stand upon. Also, your gym-weight would weigh (I don't want to calculate how much), but near the surface of a black hole gravity is so high that a feather would weigh probably thousands maybe hundreds of thousands of tons in your scenario. Lifting with no surface below you and feathers weighing tons poses several problems with your scenario. I'm not sure any of that applies to your question in your thought experiment, I'm just pointing out some of the difficulties that arise.
@PM2Ring points out that if the black hole is large enough the weight and surface gravity does decrease, which sounds odd, but seems correct, but no black holes are that large, and there's still the no surface to stand on.
Now, I’m hovering one meter below the event horizon. Would I not be
able to lift the weight up one meter at speed less than escape
velocity (which at this point is speed of light plus y; not sure if x
and y are equal), or would I not be able to hover at this position?
You can't hover below the surface of a black hole unless you have an energy source that defies the laws of physics.
Perhaps it's easier to think of in terms of space-time, which I'm not an expert with, but the definition of space-time inside a black hole is such that no amount of acceleration can move any thing further away from the singularity. No fuel, no manual force can move any part of anything in a direction away from the center of the black hole. Every direction points to the center. Space inside a black hole becomes like like time. You can slow time down but you can't reverse it. Lifting something upwards is no more possible than stepping backwards in time.
I don't know if that helps. Maybe someone can explain it better than me.