Learn English – “hearing him speak” or “hearing him speaking”

No, because all examples describe a future state. (Although in practice, neither would be viewed as inappropriate.**)

All of those examples apply to some possible future event (that, as you say, the speaker is thinking of doing or would like to do, i.e., hearing news, hearing progress, hearing proposals), so the formations as given can be correct.

If they were phrased in the past tense, interested to would be appropriate. For example, if the speaker had already heard the proposals, "I was interested to hear any proposals he might have had" would be a proper statement of his mood after hearing the proposals.

** The speaker could also be considered to be reflecting on a hypothetical future state, so "I would be interested to hear any proposals he might have," meaning "If I did hear any of his proposals, 'interested' would describe my mood" wouldn't come across as incorrect.

Both sentences are correct — they simply use different grammatical forms as subjects:

Like you said, the first sentence uses the gerund "hearing" as its subject. Turning it into "The hearing of" is basically the same thing, but it can make it confusing as to whether the hearing belongs to "this" (a possessive use of "of," which changes the sentence's meaning) or whether its an objective use of "of" — identical in meaning to "Hearing this..."

The second sentence uses a subjective infinitive, which is identical in meaning to the gerund example but (as FumbleFingers suggested) can be more limited in use.