Related question: Are manifold subsets submanifolds?
Assume all manifolds, topological or smooth discussed here have dimensions and do not have boundary.
Let $A'$ and $B'$ be sets with $A' \subseteq B'$.
Question A: Are these correct?
As far as I know, all sets can be made into topological spaces.
By (1), make the sets $A'$ and $B'$ into, respectively, the topological spaces $A$ and $B$.
I haven't thought about whether there are some sets that cannot be given topological spaces that enable them to become smooth or topological manifolds, but as far as I know, some topological spaces cannot be made into smooth manifolds or even topological manifolds...such as ones that aren't Hausdorff I guess.
By (3), assume $A$ and $B$ from (2) can be made into smooth manifolds $(A,\mathscr A)$ and $(B,\mathscr B)$ where $\mathscr A$ and $\mathscr B$ are smooth atlases.
By (4) and the above related question, $(A,\mathscr A)$ is not necessarily a (regular/an embedded) smooth submanifold of $(B,\mathscr B)$ or even an immersed smooth submanifold.
Question B: Does there exist a smooth atlas $\mathcal A$ where $(A,\mathcal A)$ becomes a smooth submanifold of $(B,\mathscr B)$?
I hope Question $B$ is equivalent to both
(C) "If a topological subspace can become a smooth manifold, then can it become a smooth submanifold?"
(D) "Can smooth manifold subsets $(N,\mathscr N)$ of smooth manifolds $(M, \mathscr M)$ always be made into smooth submanifolds $(N,\mathcal N)$ of $(M, \mathscr M)$?"
If one of (1)-(5) is wrong, then $B$, $C$ or $D$ could be meaningless or, if meaningful, not equivalent to the other meaningful ones. Please answer the meaningful ones among (B),(C) and (D), and please point out which are equivalent or not.
Not concerned about uniqueness at this point. You can say something about uniqueness if you want.