So, if all the bodies are embedded in space-time and moves through it, is there some kind of 'friction' with space time of the planets? For example, the Earth suffers friction when moving near the sun due the curvature and General Relativity and loses energy?
If a planet loses energy due to friction can this energy loss be measured?
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?
is there some kind of 'friction' with space time of the planets?
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:
if a planet loses energy due to friction can this energy loss be measured?
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).
Besides the visible matter in space, space is "full of" electromagnetic radiation (EMR). However, since the density of the EMR is very small, the density of a planet very large, and the charge of the planet essentially neutral, there is an extremely small "interaction" between a planet and EMR. It takes a very massive object to generate barely detectable "friction waves" - in space.
We cannot gain an understanding of the nature of Einsteinian spacetime at a planetary scale. All matter, including planets, are composed of particles, and to answer this question we should consider the link between spacetime and particles, such as quarks.
Clearly, a particle has mass, and thus inertia. But what is inertia? It is presumably what the questioner means when he talks of 'friction'. A particle has inertia, or friction if you like, because it is bound to the spacetime field postulated by James Clark Maxwell in the 19th century, in Maxwell's equations.
Modern theory looks upon inertia as being due to a coupling or bond, which attaches a quark to the spacetime field. So yes, all particles have inertia, and the inertia is the reason why a particle has mass: mass being merely a measurement of the amount of energy which is required to break the coupling bond.
