Space-time, as you said curves around a body in space, and the greater the mass of the object the greater the curvature, this, in the simplest way can be described using light cones and Schwarzschild.
Black-holes create many odd situations in space in which odd events take place. Black holes, although often though of as places in themselves, they have their own geography. There are two checkpoints to look for when one falls in a black hole these are called, the Horizon and the Singularity. The Singularity is the easiest to explain, in terms of description.
As you can see the singularity is the center point of the black hole, it also shows us the relative place of the Horizon, our second checkpoint. The Horizon is defined by the size of the black hole. The calculation in how to work out the Schwarzschild radius, (where to place the Horizon) is stated below:
The important thing about the Horizon is that NOTHING SPECIAL happens if one were to pass it, on e would feel no special sensation and may not even know they had passed the horizon unless they had a way of measuring it. The only interesting thing that happens at the Horizon is that when someone else watch one they would see an odd situation, but that would be going on to a completely different discussion point. The Singularity is the only place in which one would feel anything, and that thing would a infinite tidal forces, ripping one apart because gravity would be infinite. This is where we come back to your original question, gravity is NOT a force, it is only talked of as a force so that it is simple to imagine, but a curvature of space, and therefore curvature would be infinite. This is only at the singularity though, at the horizon the curvature would not be quite as intense. If you want to know the exact equation then look up Ricci/Riemann Curvature equations as I have chosen to describe this is in a non-mathematical sense. Below I have left a diagram of a black hole to help you visualize it.
(original source)
I hope I helped and did not ramble too much. If you found my explanation helpful and not too boring then look up more and even try listening to Leonard Susskind, or Brian Greene, both really interesting and have videos on YouTube. It is really interesting so dive as deep as you can, then go even deeper!
Best Answer
I think the question suggests you are thinking of space-time as if it were e.g. a substance, like a fluid, that we move through. That's not how we view space-time, at least in pure general relativity.
But the question you ask is a deceptively simple one and it raises some complex questions. And I don't think we actually can answer them exactly because I'm not sure we have a definitive answer to the most basic question hidden in your answer: What is space-time?
There is a "kind" of friction, but perhaps "interaction" would be a better choice of word, as I'd prefer to avoid the notion of classical friction forces.
We say that when an object moves through space time it distorts space time - stretches it, compresses it. Mass creates distortions we describe as gravity.
It's a little deeper than that.
We also know, thanks to the wonderful LIGO experiments, that these gravitational effects do distort space in a wave-like way. And an object can lose energy (has to, in fact) when it creates such waves.
Which leads us to this:
No (I suppose I should say, not at our technological level). It's tiny.
The gravitational waves we have measured (which represent the closest thing to your friction loss) are due to the collisions of huge black holes, and the disturbance they make is so small that LIGO scientists are pushing the boundaries of measurement to detect them at all. A planet is a tiny thing compared to those black holes and it barely makes a dent, as it were, in space time by comparison.
But it's worth saying that our current understanding of space-time is a little basic. We don't have a clear idea of how the quantum world fits into the grand scale of relativistic space-time. At present we have two models, one of a small scale space-time filled with a sea of virtual particles and the other of a pure, clean empty space time with the odd idealized gravitational mass in it. We don't have a single theory connecting them, so we don't really have a proper theory of space-time (or perhaps something deeper than that is needed - no one knows).