I think that this is pretty field dependent. The first two sentences don't strike me as odd, though J-Kun strongly disagrees in the comments. In particular, I would say that the phrase "in this paper we will show" is common in the papers I read. I have first hand experience that the future tense used in this way is common in mathematics, computer science, and philosophy.
I also have very strong opinions about the fact that papers should not be dry. The fact that some researchers praise dry writing is fundamentally misguided in my mind. Papers are written by humans to be read by humans, and should be enjoyable to read. In my field (theoretical computer science), historical anecdotes, philosophical interludes, jokes, and amusing names are commonplace in academic writing. I strongly discourage you from aiming for dryness in any portion of your paper.
Given the field-dependent nature of stylistic issues like this, consulting the editorial guidelines of the journal or journals you hope to publish this paper in / have published in in the past would likely be your best option. Even if they don’t have specific style guidelines, you can usually in for what is preferred by reading other papers published by these journals.
EDIT: In your edit, you specify that you're in computer science. In my experience, computer science writing is far more conversational, and computer science culture is far more casual, than most other fields. I doubt anyone would bat an eye at either of the first two phrases. In fact, the paper I currently have open on the Arthur-Merlin Protocols says:
In Section 3 we will also use the representation of Boolean functions by polynomials with real polynomials (p. 5)
We will show that this
protocol needs communication at least Ω(√na), which implies the theorem. (p6)
and has similar constructions in at least four other places. IIRC, Babai's original paper introducing the A-M protocols had the first four pages entirely in the form of a faux mythological story about King Arthur.