The first thing to note is that both the baker and cupbearer would be highly trusted and influential officials in Pharaoh's court. This might not be immediately obvious, since being a cupbearer (literally, someone who carries a cup full of wine to the king) sounds very menial, and almost like a slave's role. Being a baker is not much above that, because you just cook bread.
However, the situation is different than that. The cupbearer is responsible for ensuring that the wine is not poisoned, and the baker is responsible for ensuring the bread is not poisoned. Therefore they are directly responsible for protecting the king's life, and conversely they are positioned so that they could kill the king if they desired. There are records of this happening many times to ancient sovereigns, so they will naturally be extremely judicious in their selection of cupbearers. Furthermore, the cupbearer was a sort of confidante or advisor to the king. As such, he would have tremendous influence over the king.
I am able to find resources discussing the importance of the baker. However, due to the symmetry of the passage it seems to be implied that the cupbearer and the baker would be roughly interchangeable roles in terms of significance. From what I can tell, the baker would possibly not have the same advisory role, but would nevertheless have to be highly trusted because of the risk of poisoning.
While it doesn't say why Pharaoh is angry with his servants, it is possible they were accused of a plot to poison his wine or bread, in accordance with what I have previously stated. This might be the most likely scenario, but it is still speculative.
There have been several dreams in the bible already. We saw Abraham have a dream (making the covenant with God), Jacob had a dream (the ladder with angels), and Joseph had two dreams earlier. Each time a dream has been recorded, I offered some ideas on how the dreams can be interpreted and their significance. This is almost premature because I have no explained *why* dreams can be interpreted, and how the symbolism should be rendered. I am not going to explain this here either (if you are very curious, you can look at my "how I categorize dreams" for some general ideas around categorizing dreams which is applicable to biblical dreams as well, although unfortunately I don't say much about dream interpretation). What this passage can teach us is that dreams *have* interpretations, and that those interpretations belong to God as Joseph says.
If interpretations belong to God, then how is Joseph sharing the interpretation? The answer is simple: he is filled with the Spirit of God through his connection to the Abrahamic promise and the Spirit gives him the interpretation. This is really amazing because God never said anything about giving his Spirit to people or interpreting dreams in the actual Abrahamic covenant or blessing. It just says stuff about having many descendants and being blessed (largely in a material sense). Joseph's gift of dream interpretation (coupled with "interpretations belong to God") is then very surprising and illuminating. It shows us that the Abrahamic covenant is deeper than it first appeared in Genesis 12 when we first saw it, or Genesis 15 when it was expanded or Genesis 22 when it was re-confirmed and established. In all of those cases, the promise was tightly focused on 1) many children, 2) inheritance of land, 3) "blessing" and "greatness", which are ambiguous and undefined, but probably related to points 1) and 2). Now we are seeing this promise surprisingly expanded to include wisdom and dream interpretations, or in the broadest possible sense, "access to God's interpretations of dreams". Note that there was no suggested interpretation for either of Abraham's or Jacob's dreams. In both of those cases, they are simply recorded and left without explanation. Perhaps we will discover that the Abrahamic blessing is richer and deeper than it first appeared.
The two dreams are probably reflecting the two dreams of Joseph earlier in this story. However, while Joseph's two dreams matched each other and showed agreement and finality, these two dreams are opposites of each other. Since they share symbolism (the number 3), they mirror each other to highlight their differences. This is how Joseph's subsequent interpretation fits the content of the dreams pretty well.
This chapter uses a pun on the expression "lift up the head of", which is used to mean both "restore to honor" and "execute". The NIV has a good translation of this: "Pharaoh lifted up the head of the cupbearer, but lifted off (i.e. removed) the head of the baker". The NLT translation is also moderately good; I am not a fan of the Message translation of this passage because it does not honor the pun in its translation.
This is one of those areas that highlights the difficulty of translating the bible, because you want to be able to capture the original author's cleverness in this passage, yet the double entendre is an idiomatic expression that is not commonly used in modern english, "to lift up head." As such, a word-for-word translation would be confusing since the idiom is unfamiliar, but a thought-for-thought translation would likely strip out the idiom in order to capture the meaning, but would lose the pun. That's why I like the NIV approach, which mostly maintains the original sense, but changes a few words to make it clearer what is being talked about.
The former sense of the phrase "to lift up a head", exaltation, is extremely common in the bible and one presumes in ancient Hebrew in general. This implies that the latter sense, execution, would be the clever or unusual meaning. "To lift up a head"-meaning-execution is almost never used in the bible that I can think of, though other expressions referring to decapitation do occur fairly often. We will see the phrase "To lift up a head"-meaning-exaltation often in the Psalms in particular, as well as the prophetic literature. The OT also frequently speaks of "exalting