Is it possible that black holes trap their own Hawking radiation?
Certainly, this is precisely what happens. In a sense, only a tiny fraction of photons created near the black hole horizon escapes to infinity, the rest falls back into the black hole.
The relevant concept is black hole's greybody factors which is the ratio of flux at infinity to the flux at the horizon (for a given species, frequency, orbital angular momentum number etc. of Hawking radiation). For many modes of radiation those quantities are exponentially small, meaning that there is only a small chance that a photon with corresponding quantum numbers escapes to infinity. In fact, black holes are such long lived objects because they capture most of photons that (at least temporarily) escape the horizon.
While it is often emphasized that a photon in curved spacetime is an observer dependent concept, those photons near the black hole that do not escape to infinity should be treated as perfectly real objects. In particular, the energy of those photons contributes to $\langle T_{ab}\rangle$, averaged stress-energy tensor of quantum fields, and if we are willing to lower a detector toward near-horizon region those photons could be absorbed and their energy extracted.
Imagine the following situation: a short distance away from the black hole a perfectly absorbing screen is placed capturing all the radiation from the horizon that got to this distance and then the absorbed energy is removed by some mechanism such as thermal conduction or coolant circulation. In this case the mass loss of black hole can be considerably enhanced and its lifetime would be much shorter, see this answer for more details and further references (the image above is taken from one of the references).
… could there be black holes with extremely large angular momentum that could transfer themselves part of it to escaping photons (even if they initially had small amounts of angular momentum upon escaping)?
It doesn't work like this. The component of photon's angular momentum along the black hole axis of rotation is a conserved quantity of photon's motion, so a photon once created will not take any of the black hole's angular momentum. Instead, there is a superradiance phenomenon: a wave scattering of the black hole is amplified under certain conditions, or in quantum language, occupancy numbers of a EM wave (or other types of boson quanta) are increased upon scattering i.e. new photons are created in this radiation mode. One can think of superradiance as a stimulated emission, whereas Hawking radiation is spontaneous emission, those two effects are closely related (similarly to relations between Einstein's A and B coefficients).
By itself superradiance does not provide a mechanism to confine the radiation, yet if there is some such mechanism then superradiance can extract considerable quantities of black hole angular momentum and rotational energy in a short time. One such mechanism is known as black hole bomb, hypothetical device where a mirror is placed around a rotating black hole extracting huge amounts of rotational energy via EM radiation. Another mechanism are massive bosonic fields which can have stable(-ish) bound modes around black holes (a sort of gravitational atom). If such modes exhibit superradiance their occupancy numbers would be growing exponentially until black hole loses enough angular momentum. Of course, for this last scenario to work our Universe must have stable weakly interacting bosons with small masses (for best efficiency Compton wavelength of such boson must be comparable to black hole's Schwarzschild radius). Potentially, this could lead to observable consequences such as gravitational wave signatures and populations of BHs without high spin parameters. Alternatively, observation of BHs with large angular momentum provides constraints on possible existence of light bosons.
