I too am struggling with this and I think it is because of confusion between adjectives and nouns in the instructions / translations. For example, you say that Sentir is often followed by a noun, and yet you use adjectives in the translation of the sentences i.e. "siento pena" translates as "I feel sorry" - but "sorry" is an adjective not a noun. The noun of sorry is "sorrow." Hence "I feel sorrow" would be the correct translation if specifying the use of a noun. A second example is "sentimos mucha alegría' which you translate as "we feel very happy" but "happy" is an adjective. The noun of happy is "happiness" so "I feel happiness" would be the translation of the noun form. I completely get how these translations of the noun form would be very clunky, but I think it may help to point this out.
The issue may be - but you don't state it, that Sentir appears to be used to express emotional feelings or something that is sensed physically, and emotions are mostly expressed in the adjective form in English "I feel sad because my cat died" or "I feel delighted since my partner left me." Both adjectives are describing how I feel. If the noun forms "sadness and delight" were used, it would describe what I am feeling. Your instructions say that "how" you feel takes the reflexive form. This seems to contradict the fact that we feel feelings, and that is how we feel when we are feeling them.
I hope this makes sense.