From this question and the comments, we can relax $F$ from diffeomorphism to local diffeomorphism. My question is further relaxing from local diffeomorphism to immersion.
Is Part A (the inner product part) satisfied?
I was able to show that that $F$ is an immersion at $p$ is equivalent to saying that if $\langle X_p, X_p \rangle'_p = 0$, then $X_p=0_p$, the zero tangent vector in $T_pN$. This is one-third of the positive-definiteness part of the inner product part.
I think the other two-thirds of the positive-definiteness part, the symmetry part and bilinearity part are satisified without assuming $F$ immersion.
As for Part B (the smooth part), I'm going to use Exercise 1.5 and its proof. Is this correct?
2.1. Let $X,Y \in \mathfrak X(N)$ and $p \in N$. By Exercise 1.5 and its proof, we can write $\langle X, Y\rangle'$ locally as $(\langle X, Y\rangle')_p = (\langle X, Y\rangle'|_{U_p})_p = (\langle A,B \rangle \circ F)(p)$, through writing $\langle X, Y\rangle'|_{U_p} = \langle A,B \rangle \circ F$, for some $A,B \in \mathfrak X(M)$ where $U_p$ is a neighborhood of $p$ in $N$ and $A_{F(p)} = F_{*,p} X_p$ and $B_{F(p)} = F_{*,p} Y_p$.
2.2. By assumed smoothness of $\langle, \rangle$, we have $\langle A, B\rangle \in C^{\infty}M$.
2.3. Therefore, by (2.1) and (2.2), each $\langle X, Y\rangle'|_{U_p}$ is smooth as a composition of smooth maps $F$ and $\langle A, B\rangle$.
2.4. A map $G: P \to Q$ of smooth manifolds is smooth if each $r \in P$ has a neighborhood $V_r$ in $P$ such that $G: V_r \to Q$ is smooth.
2.5. Therefore, by (2.3) and (2.4), the entire $\langle X, Y\rangle'$ is smooth.
Update 1: It appears this is true, is called "pullback metric" and is what Paulo Mourão proved in the aforementioned question. Therefore, I hope this is not a duplicate because I gave my own proof (at least for Part B).
Update 2: It appears I don't assume $F$ immersion for Part B. Or do I?
