Completeness, compactness and Löwenheim–Skolem

Which statement is Gödel's completeness theorem (as opposed to soundness, or to the incompleteness theorem)?

Which of these classes of structures are first-order axiomatisable (by a possibly infinite set of sentences)? Select all.

Colour the numbers $0, 1, \dots, 9$ with $3$ colours so that numbers differing by $1$ or by $2$ get different colours. How many such colourings are there?

In the Henkin proof, why do we add a witness constant $c_\varphi$ and the axiom $\exists x\, \varphi(x) \to \varphi(c_\varphi)$ for every formula $\varphi$?

Suppose $T$ axiomatised the torsion groups, and let $\Gamma = T \cup \{c^n \neq e \mid n \ge 1\}$. Why is every finite subset of $\Gamma$ satisfiable?

Let $M \equiv (\N, 0, 1, +, \times, <)$ contain an element $c$ larger than every numeral. Which statements are true? Select all.

A countable model $(M, E)$ of ZFC contains an element $p$ that $M$ believes is $\mathcal P(\omega)$, and $M \models$ "$p$ is uncountable". Yet $p$ has only countably many $E$-members. What is the correct resolution?

Mathlib defines T.IsFinitelySatisfiable as ∀ T0 : Finset L.Sentence, ↑T0 ⊆ T → (↑T0 : L.Theory).IsSatisfiable. Which Lean statement is the compactness theorem?

Numbers $i, j$ are joined when $|i - j| \in D$. Write first_empty_level(k, D, max_n) returning the least $n \le$ max_n such that $\{0, \dots, n - 1\}$ has no proper $k$-colouring, or None if every $n \le$ max_n has one. (Build the colourings level by level: each proper colouring of $\{0, \dots, n - 1\}$ is extended by a colour for $n$ that differs from the colours of $n - d$, $d \in D$.)