When Does an Atlas Uniquely Define a Manifold?

I am totally new to differential geometry and am having trouble understanding a very basic idea. In what follows, I apologize for being gratuitously pedantic, but I want to be sure I clearly understand what's going on.

If $M$ is a set and $T$ is a topology on $M$ such that $(M,T)$ is Hausdorff and second countable, then $M$ is a topological manifold if for all $p\in M$ there exists an ordered pair $(U,x)$ such that $U \subset M$ is $T$-open and $x:U\rightarrow \mathbb{R}^d$ is a homeomorphism whose image is an open subset of $\mathbb{R}^d$ in the standard topology.

Ordered pairs $(U,x)$ that satisfy the conditions in the above paragraph are called charts on the manifold. An atlas for $M$ is a collection of charts on $M$, $A = \{(U_a,x_a)\colon a \in I\}$, such that $\cup_{\alpha\in I}U_a = M$.

Question 1: Does every manifold have at least one atlas?

My answer: I believe so, since by the definition of a manifold there exists at least one chart for each point, and the collection of either all or at least one of the charts at each point can be taken as an atlas. Perhaps however there is some technical problem in set theory with this construction.

Question 2: Does an atlas uniquely define a manifold? That is, if $A$ and $A'$ are atlases and $A \neq A'$, is it necessary true that the manifolds with $(X,T)$ as their underlying space but with atlases $A$ and $A'$ respectively are different? (In the naive sense--not considering the possibility that they are diffeomorphic)

I believe the core concept I'm struggling with here is what the naive notion of equivalence is for manifolds. (For example, for topological spaces "naive equivalence" means that the two underlying sets are equal and the two topologies have exactly the same open sets, rather than the existence of a homeomorphism, which is a more sophisticated notion of equivalence.)

If instead we define a topological manifold as an ordered triple $(M,T,A)$, where $A$ is an atlas, my confusion vanishes. But then naive equivalence requires exactly the same charts in the atlas, which might be too much to reasonably say that two manifolds are the same. I've also not seen this definition in any of the references I'm using. This brings up the following question.

Question 3: Is it possible to define a manifold as an ordered triple, as in the paragraph above?

from Hot Weekly Questions - Mathematics Stack Exchange