This is about Silverman's AECs book.
The author says at the beginning of Chapter $3$ that he's going to assume field $K$ to be algebraically closed. Then later says that every elliptic curve defined over $K$ is isomorphic to a smooth cubic curve in $\mathbb{P}^2$, that is, it has a Weierstrass form $$ y^2 +a_1xy+a_3y = x^3+ a_2 x^2 +a_4 x+ a_6.$$ where $a_i \in K$ for each $i$.
The proof of this fact uses an application of Riemann-Roch Theorem on the curve $$y^2 = (x-e_1)(x-e_2)(x-e_3)$$ with $e_i \in \bar{K}$, combined with the fact that for a curve defined over a field $K$, basis of the Riemann-Roch space of every divisor defined over $K$ are functions in the function field $K(C)$. All these results are given in Chapter $2$ which assumes at the beginning that $K$ is a perfect field.
Q.1 If we are already assuming $K$ to be algebraically closed, there is no need to confuse readers and say $E$ is defined over $K$ because that is the same as saying that it is defined over $\bar{K}$, right? Maybe I'm missing something but that doesn't seem right. Also, later in the Chapter $3$ author frequently consider the Galois group, $G_{\bar{K}/K}$ but that'll be trivial in that case.
Considering $K$ to be an arbitrary field seems to make more sense. But then the author uses some results from Chapter $2$ which assumes $K$ to be perfect so maybe the correct assumption for Chapter $3$ onwards is, $K$ is an arbitrary perfect field unless mentioned otherwise, right? (Later chapters on finite, local, and number fields also make sense in that case.)
Q.2 Why is it needed that $K$ must be a perfect field in Chapter $2$? I know that because we are ultimately interested in the algebraic extensions of $\mathbb{Q}$, $\mathbb{Q}_p$ and $\mathbb{F}_p$, so this doesn't really matter but why even assume so in the first place? Can all the discussion in Chapter $2, 3$ hold for arbitrary fields as well? (I suppose I'll have to inspect each result there)
The author does not state at the start of the chapter that $K$ will be taken to algebraically closed. He only writes that the book will start by studying elliptic curves over the algebraic closure since that is where a lot of the "geometry" lives. Then one can apply results from Chapters 1 and 2 by passing to $\bar K$. Any property or related object that doesn't depend on $K$ is amenable to study from this viewpoint, and then have applications to the arithmetic of the curve.
For instance, the Tate module is defined by putting together all of the $\ell$-torsion on the elliptic curve instead of taking just the $\ell$-torsion in $E(K)$. Studying the action of Galois on the Tate module is one of the ways to translate data from the algebraic closure back to the ground field.
As far as being perfect, without that assumption many of the results in Chapters 1 and 2 are no longer true, and right at the start of Chapter 1 it is stated that $K$ will be taken to be a perfect field throughout the book. Relaxing the assumption that the field is perfect can probably done in some places, at least at the cost of adding extra hypotheses to various theorems. I think it would make the exposition a lot more technical. Of course people can still study elliptic curves in those settings (e.g. over function fields).