Set membership and inclusion confusion
You're correct that 1 and 4 are true.
On the other hand, if we're doing Zermelo-Fraenkel set theory, then the truthvalues of 2 and 3 cannot be determined from the information given. For example, 2 says that $\{1\} \in A$, or in other words that $\{1\} = 1$ or $\{1\} = 2$ or $\{1\} = 3$ or $\{1\} = 4$ or $\{1\} = 5$. Are any of those statements true? Well they could be. In Zermelo's approach to defining the natural numbers, $\{1\} = 2$ is true, indeed this is the definition of $2$. However, in von Neumann's approach, the set $\{1\}$ is not a natural number and thus not an element of $A$.
Similarly, the truthvalue of 3 cannot be determined. It all comes down to how you define the notion "natural number."
On the other hand, other approaches to the foundations would deem 2 and 3 "ill-formed" and therefore nonsensical. They're not even false; they're just nonsense.
To answer your questions: Membership can be a relation between two sets (at least in ZFC). For example, $\mathbb{R} \in \{\mathbb{R}\}$ is true. Furthermore, there is no distinction between "elements" and "sets" in ZFC; everything is a set, and "element" is a relationship: we can say the set $\mathbb{R}$ is an element of $\{\mathbb{R}\}$, but there's no point in saying that something "is" an element, period.
This was quite a rushed answer because I need to get back to my assignments, so please comment with any questions you may have.
Sets can be members of other sets. Element is just a term for a member of a set, it is not synonymous with "not a set". For example, given a set $A$, the power set of $A$ denoted by $\mathcal P(A)$ is a set whose elements are sets.
Numbers themselves can be represented as sets, and in modern set theory they are, and in fact everything is a set in the context of modern set theory. However if you are doing naive set theory then it is possible that you are assuming that numbers (say up to $\Bbb R$ or so) are not actually sets, but rather some atomic entity.
If this is indeed the case, then writing $1\subseteq A$ is indeed meaningless, because $1$ is not a set; however in the former case it is meaningful as you interpret $1$ as a particular set (e.g. $\{\varnothing\}$ is a common way, as $\varnothing$ is often represents $0$). If indeed $1$ is a set in your context, you need to know what is that set and then you can decide whether or not $1\subseteq A$ is true or false.