Is it possible that when matter and energy enter the singularity inside a black hole, that it all re-appears at the big bang? Some say that black holes will eventually swallow all the matter in the universe, so if it appears at the big bang, the universe will thus recycle itself.....
The matter, from the point of view of the external observer, never even reaches the event horizon. Watch your hypothetical astronaut fall towards it, with your hypothetically perfect telescope, and the astronaut's watch just clicks slower....and slower........and slower...................
You get the idea.
From the point of view of the astronaut, they're spaghettified (sooner or later, depending on the size of the black hole) - and this is the crux of the matter, and your question.
It's relatively well accepted that black holes do, eventually, evaporate due to Hawking radiation. This generally takes a Very Long Time though...and the bigger the black hole, the longer. Anything astronomically observable to us probably has an evaporation time of many times the age of the universe, in the hundred of billions of years and upwards.
But they do evaporate. And the mass-energy of everything that fell in, comes back out. The big question is whether there's any information preserved - does the astronaut's mass come out as anything other than (by definition) unencoded "junk" data in the form of black body / thermal radiation, or is there any trace of the original structure?
It's likely (and a tad depressing) that it's the former, the astronaut simply becomes a bunch of randomly dispersed photons....but it's still an open question.
Loosely speaking, it becomes part of the black hole, adding to the mass of the black hole. Exactly what goes on near the singularity is not well understood. It's an area where space time curvature is extremely high (General Relativity is needed), but also very very small (Quantum effects dominate), however GR and Quantum mechanics don't get along very well. Understanding that will mean finding a theory that describes how gravity works at a quantum level which is an active and major area of research.
However Hawking radiation is a thing (yes as in first name Stephen), and over very very long time spans a black hole will evaporate due to it. Nothing escapes from inside the event horizon mind you. Rather quantum effects right at the event horizon. The basic example usually given is that of a particle-antiparticle pair being created due to the gravitational energy of the black hole and right at the event horizon of the black hole. It's possible for on particle to fall into the black hole, while the other escapes, however since the energy to create that particle came from the gravitational energy of the black hole, the mass of the black hole must be reduced by a very very tiny amount. For most black holes this mass lost is far far less than the mass they gain from sources like the CMB. Under that critical mass however and they'll evaporate. The end of that process is quite quick (a smaller black hole loses mass to this process much much quicker than a large one) and releases a lot of energy at once; we should see this eventually as gamma ray bursts.
Given a sufficiently long time however as the energy density of things like the CMB decrease, and that critical mass needed to remain stable will increase and they'll evaporate and vanish. For a solar mass black hole that will take something on the order of an unvingtillion years (that's a 1 followed by 66 zeros) many many many many times longer than the current age of the universe. For the largest supermassive black holes this will occur somewhere on the order of duotrigintillion years (that is, a 1 followed by 99 zeros).
The mass/energy is consumed behind event horizon. No information can be retrieved from behind EH, so we do not know whet goes on in there in practical, except only theoretically. Theory says it meets the singularity. Hawking's radiation is supposed to be results of quantum fluctuations right on the event horizon.
As far as cyclic universe - it is more likely a cyclic universe than other speculations.
The chance that it is just a one time universe (big rip, or heat death), is very low just because the universe exists right now.
If it was not cyclic, it would have already done with big rip or heat death. What would stop it from not doing so already. Why it started exactly when it started and not before?
If a one time universe can just pop into existence, then another one time universe can pop into existence after this one is dead. That also makes it cyclic in some sense.
Chance that the universe pops into existence only once, and dies (in big rip, or heat death), and then never pops into existence again, makes the first time itself highly unlikely. The reasons that caused existence first time, where did they go?