How do we view natural transformations as functions 1.The definition asserts that natural transformation is a map of two functors. However, from the definition, given tow functors $F,G:C,D$, we associate every element $x$ in $Obj(C)$ a morphism $F(x) \to G(x)$. Isn't this a function from $Obj(C) \to Mor(D)$? In what sense is it a map of two functors?
2.We have the identity natural transformation defined as sending every $x$ to the identity morphism. Why? How do we define the composite of two natural transformations and the inverse of a natural transformation?
 A: The statement that a natural transformation is a map between functors is wrong, simple because functors are not sets (and even if they were realised as sets, it is certainly not those sets that the natural transformation maps between). But natural transformations can be thought of as morphisms between functors; thus one can define a category whose objects are the functors $C\to D$, and whose morphisms are natural transformations between such functors.
Since they are part of the definition of this category, the statement that they are morphisms is kind of bland (we just define morphisms to be natural transformations); however it does come with an obligation: one must be able to define composition of natural transformations, and have the properties one requires for composition of morphisms. Defining composition is easy: if $F_1,F_2,F_3: C\to D$ are functors, and $\eta$ is a natural transformation form $F_1$ to $F_2$  and $\theta$ is a natural transformation form $F_2$ to $F_3$, then the composite transformation $\theta\circ\eta$ should define for every object $c$ of $C$ a morphism $F_1(c)\to F_3(c)$, which can obviously be taken to be the composite (in the category $D$) of the morphisms $\eta(c):F_1(c)\to F_2(c)$ and $\theta(c):F_2(c)\to F_3(c)$. This composition clearly is a natural transformation $F_1\to F_3$, and required (partial) associativity of this composition is ensured by the same property in the category $D$. Composition also needs an identity at each object (i.e., at each functor $F:C\to D$), and this is where the identity natural transformation comes in. Clearly is should be such that it associates to every $c$ the identity morphism of $F(c)$, or else it would not be the identity for composition of natural transformations.
A: A natural transformation $\eta : F \to G$ between two functors $F,G : \mathbf{C} \to \mathbf{D}$ is an $\operatorname{Ob}(\mathbf{C})$-indexed family $\left(\eta_{c}\right)_{c \in \mathbf{C}}$ of morphisms $\eta_c : F(c) \to G(c)$ (necessarily in $\mathbf{D}$), such that the naturality squares commute.
To answer your first question, note that to index a sequence of elements from a set $S$ with indices $I$ is the same thing as a function $I \to S$. So you could view a natural transformation as a function $\operatorname{Ob}(\mathbf{C}) \overset{\eta}{\to} \operatorname{Mor}(\mathbf{D})$ satisfying the necessary conditions (that the domain and codomain of $\eta(c)$ are $F(c)$ and $D(c)$, and that the naturality squares commute). But this is only a matter of terminology.
To answer your second question, remember what an identity morphism in a category does (given the notion of a natural transformation, we ought to expect there to be a category of functors $[\mathbf{C}, \mathbf{D}]$): it's the left- and right-identity in the composition algebra of morphisms. Since natural transformations are composed pointwise, having the identity morphism at each component is the obvious (and only) choice.
A: *

*You can think of it as a function which indexes morphisms in the category $\mathcal{D}$ with the indexing  set being objects in $\mathcal{C}$, i.e. we have that a natural transformation $\eta: Ob(\mathcal{C})\to Arr(\mathcal{D})$ such that $\eta (x)= \eta_x:F(x)\to G(x)$ also written as $\eta(x)\in$ Hom$_{\mathcal{D}}(F(x), G(x))$. Here we can see that the image of $\eta$ is the map between the images of the functors $F$ and $G$ at the object $x\in \mathcal{C}$. 


But we also impose the extra condition that for any morphism $\alpha \in$ Hom$_\mathcal{C}(a,b)$ for $a,b \in Obj(\mathcal{C})$ $G(\alpha)(\eta_a)=\eta_b ((F(\alpha))$ i.e. the diagram seen in this wikipedia article https://en.m.wikipedia.org/wiki/Natural_transformation commutes.
We can view the diagram commuting as translating between these functors i.e. $\eta_a:F(a)\to G(a)$ and $\eta_b: F(b) \to G(b)$ in the sense that $\eta$ allows us to from $F$ to $G$.


*The identity natural transformation is just the natural transformation that goes between the same functor, (instead of different functors as mentioned previously) and maps each $a\in Ob(\mathcal{C})$ to the identity map of its corresponding image under the functor $F$ i.e. the map $id_{F(a)}$.


For your remaining questions you should just look up the definitions.
