What is the method of purification (tahara) and prayer for someone suffering from second-degree internal hemorrhoids? Is it necessary to wait until the fleshy protrusions retract inward? And are they considered impure even after washing them?

The obligation in istinja (definition)' (cleaning oneself after relieving oneself) with water is for the area to return to its rough state, as it was before passing waste. Overriding doubt (ghalabat al-ẓann) is sufficient for this. If you wash the area with water, you must wash what is considered outwardly visible, including any protruding appendages, if you perform istinja' before they return to their place. If you are predominantly certain of the purity of the area, then the obligatory istinja' has been achieved, whether those appendages return inward or remain protruding. You are not required to repeat istinja' unless you are certain that something necessitating it has exited.