Exegeting the passage
τῇ δεξιᾷ οὖν τοῦ Θεοῦ ὑψωθεὶς τήν τε ἐπαγγελίαν τοῦ Πνεύματος τοῦ Ἁγίου λαβὼν παρὰ τοῦ Πατρὸς ἐξέχεεν τοῦτο ὃ ὑμεῖς καὶ βλέπετε καὶ ἀκούετε
So then, exalted to the right hand of God, and having received the promise of the Holy Spirit from the Father, he has poured out what you both see and hear. (Acts 2:33 NET)
Beginning with an exegesis of the passage, I would begin by dividing the text differently from how you have it above. I wouldn't connect the 'promise' of your section B) with Psalm 16, I would connect it with the surrounding phrase "the promise of the spirit holy having received from the Father". Assuming the author of Luke & Acts are the same, there's a clear parallel:
And look, I am sending you what my Father promised. But stay in the city until you have been clothed with power from on high." (Luke 24:49 NET)
The author is demonstrating a fulfilment of Jesus' earlier words, so that is a step we need to account for, before we start drawing from Prophetic literature on the topic. In Acts 2:40, "With many other words he testified and exhorted them...", the author makes it clear that his recording is a selection of Peter's words and not their entirety.
Luke 24:49 was spoken by Jesus post-resurrection to the disciples, and so the re-use of the language here would likely not be as significant to Peter's audience as it is to the author's. What the recipients of Acts can know is that this is not just "Peter's claim", but rather it's Jesus' claim which was taught to Peter previously. Whilst the words are very significant to Peter's audience if they pertain to Old Testament Prophecy (below), it is likely that they are even more significant to the recipients of Acts.
1. Do Prophecies indicate the Messiah would give gifts?
As you suggest, the only tenable passage which suggests this is as follows:
"You ascend on high, you have taken many captives. You receive tribute from men, including even sinful rebels. Indeed the LORD God lives there!" (Psalm 68:18)
As we have it in the OT, this doesn't seem to naturally talk about the Messiah giving gifts... and yet Ephesians 4:8 does use it in such a sense. I'd say it's debatable how we can legitimately use this passage, but there's lots more available in an earlier question:
How can Ephesians 4:8 be a quote/translation of Psalms 68:18?
2. Do Prophecies indicate that God would have the Messiah give - the Holy Spirit, specifically?
We need to examine prophetic passages which link to the 'pouring forth' which Peter is talking about in Acts 2:33, and verify whether or not these have clear Messianic inferences. There are three 'pouring out' prophetic passages which I see clear linkage with here:
"For I will pour water on the parched ground
and cause streams to flow on the dry land.
I will pour my spirit on your offspring
and my blessing on your children." (Isaiah 44:3 NET)
"After all of this I will pour out my Spirit on all kinds of people. Your sons and daughters will prophesy. Your elderly will have revelatory dreams; your young men will see prophetic visions." (Joel 2:23 NET)
"I will pour out on the kingship of David and the population of Jerusalem a spirit of grace and supplication so that they will look to me, the one they have pierced. They will lament for him as one laments for an only son, and there will be a bitter cry for him like the bitter cry for a firstborn." (Zechariah 12:10 NET)
Isaiah and Joel don't obviously link the 'pouring of the Spirit' to the Messiah, but Zechariah certainly does. It's arguable as to whether Isaiah 44:3 is using the 'offspring/seed/zerah' in parallel to Galatians 3:16's reading of the Messiah as Abraham's 'seed', since 43:5 and 48:19 use it in clear plurality rather than as a singular title.
There is a related passage in Ezekiel which seems to have a stronger case for it, though:
"I will give you a new heart, and I will put a new spirit within you. I will remove the heart of stone from your body and give you a heart of flesh. I will put my Spirit within you; I will take the initiative and you will obey my statutes and carefully observe my regulations." (Ezekiel 36:27-28 NET)
The first and most obvious reference of this prophecy is the restoration from Exile, which was to happen at the end of Ezekiel's generation, so we need the wider context of the passage if we're going to confirm whether or not there are Messianic expectations linked to it:
“‘My servant David will be king over them; there will be one shepherd for all of them. They will follow my regulations and carefully observe my statutes. They will live in the land I gave to my servant Jacob, in which your fathers lived; they will live in it—they and their children and their grandchildren forever. David my servant will be prince over them forever. I will make a covenant of peace with them; it will be a perpetual covenant with them. I will establish them, increase their numbers, and place my sanctuary among them forever. My dwelling place will be with them; I will be their God, and they will be my people. Then, when my sanctuary is among them forever, the nations will know that I, the Lord, sanctify Israel.’” (Ezekiel 37:24-28 NET)
These passages are bridged by the first half of Ezekiel 37, the 'Valley of Dry Bones' prophecy, which is all about breath coming into dead men to make them alive. Thus I would argue that it is natural to read these passages together as part of the larger Ezekiel 33-39 section, in which case we can use a consistent hermeneutic across the passages. Note the theme of 'they/you will obey my commandments' in Ezekiel 36:28 cf. 37:24, suggesting that this Messianic event of obedience is the same as the 'new spirit', 'my spirit' which had been prophesied earlier.
Ezekiel 39 later reaffirms the promise of the 'pouring of the Holy Spirit', but explicitly in a 'returning from Exile' context, so is arguable as to whether that supports or undermines such an interpretation.
Considering Isaiah, Joel, Zechariah and Ezekiel, yes there is a clear link between the coming Messiah and the 'pouring of the Holy Spirit'. That said, the passages don't suggest that the Messiah himself would give the Holy Spirit, but rather that God would pour it upon the Messiah and others (Zechariah 12:10 as above).