1.It is natural for us now to see the natural number $1,2,\cdots$ and the operator "+", but for me it is hard to see how we define "+", i.e. I can't see the rule of $a+b$.
2.Another question is how we define "0"
Or we define "+" first or natural number first?
On
Natural numbers ($0$ excluded) are related to counting objects, they are `defined' as the number of elements in a collection, thus $a = \#\big( A \big)$.
The operation `addition' is defined as
$$ a + b = \#\Big( A \Big) + \#\Big( B \Big) = \#\Big( A \cup B \Big) = \#\Big( A \Big) + \#\Big( B \Big) $$
for disjoint sets $A$ and $B$.
$0$ is defined as the neutral element for addition, thus an element $e$ defined such that
$$x = x + e$$
or
$$ \exists 0 \in G:\forall x \in G: x+0=x $$
As for the question - that we need $0+x=x$. No we don't need to do that. Suppose that $x+0_r=x$ and $0_l+x=x$, then we can write
$$ 0_r = 0_l + 0_r = 0_l$$
they are the same.
It is hard, and in fact impossible, to answer without more information. In particular what is your foundation in which you define the natural numbers. I'll give a few options and I'll try not to get too technical.
First of all, what are the natural numbers? It is somewhat understood in mathematics that the natural numbers are the unique model of Peano axioms which satisfy the second-order induction axiom. So if we can show that our structure has this property, then it can be called the natural numbers.
What are [some of] our options?
We can work inside set theory. Inside set theory we have access to a large number of expressible relations. One way to do things is to prove that we have some form of induction for ordinals (which are actually a generalization of the natural numbers), and then show that a certain set of ordinals along with the natural operations on the ordinals satisfy what we wanted.
In that case we define $0$ as the empty set, and $n+1=n\cup\{n\}$ (recall that $n$ is a set here). We can do this induction because we are doing our induction in the context of set theory, there's no circularity in this definition.
Addition is defined by recursion, $n+0=n$ and $n+(m+1)=(n+m)+1$. Again, there's no circularity, since the recursion is not part of the theory of the natural numbers but rather in the set theory.
We can use set theory, but we can talk about cardinals (which extend the natural numbers in a whole other way). Now we don't need recursion to define our operations. We have a definition for cardinal addition, and so on, and we simply pick a "good" definition for what it means to be finite, and we declare that the natural numbers are a set of representatives for the finite cardinals.
Okay, maybe set theory is a bit too strong, and after all, we often interpret set theory itself within the natural numbers. So this is "sort of" circular. How about second-order logic? In this case you are given $\Bbb N$ as a set, along with $\leq$ or with some other structure. But this is the true structure of the natural numbers -- even if just a part of it.
Using second-order logic we have access to induction in a way that looks quite close to what we did in the case of set theory (and Quine did call second-order logic "set theory in sheep's clothing"). But the idea is that $0$ is the minimal element of $\leq$, or some other definable property which depends on the structure you are given, then you can work your way towards defining successorship, addition and so on. Just like with the set theoretic case.
First-order logic and a partial structure. Well, this gets a bit complicated. You are given the natural numbers with some structure, and from this you can sometimes define the usual operations (for example if you are given $E(x,y)$ which means $x^y$, with $0^0=1$).
Other times, you are given part of the structure, and you provably can't define the rest. You can't define the order $\leq$ just from the successor operation. You need addition.
You are given the natural numbers, with their structure. You don't have to work anymore, this is given.
Why is this even a valid approach? Because since we talk about abstract objects, at some point you will have to take something "for granted". The natural numbers seem like a good object to take for granted, since they model something we understand quite well (for a very limited size): arithmetic, and the idea of induction which we are prone to accept due to evolutionary design as animals that look for patterns (if something kept on happening, it will keep on happening).
So the question is what is your foundational bedrock, and then the answer can be more satisfying, or less satisfying. But for the most part, we take these things for granted to exist the way they are.
We agreed on some axioms that the natural numbers satisfy, so as far as we are concerned mathematically, the way I interpret the natural numbers in my mind - even if different than how you interpret them in yours - have a few fixed properties which allow us to be sure that what other people tell us about the natural numbers is coherent with our understanding of them.