Question about renormalization parameters and counterterms in Srednicki's book

In Chapter 9 in Quantum Field Theory written by Srednicki.

This chapter discusses why $Z_{i}=1+O(g^{2})$ and $Y=O(g)$, given specific values of $m$, $g$, and normalization conditions
$$langle k|phi(x)|0rangle =e^{-ikx}qquadtext{and}qquad langle 0|phi(x)|0rangle =0,tag{9.2}$$
when considering an interacting quantum field with Lagrange density:
$${cal L}=-frac{1}{2}Z_{phi}partial^{mu}phipartial_{mu}phi-frac{1}{2}Z_{m}m^{2}phi^{2}+frac{1}{6}Z_{g}gphi^{3}+Yphi.tag{9.1}$$

Denote
$$Z(J)equivlangle 0|0rangle_{J}=int {cal D}phi, e^{iint d^{4}x,({cal L}_{0}+{cal L}_{1}+Jphi)}tag{9.5}$$

where $J$ is an external field source, and

$${cal L}_{0}=-frac{1}{2}partial^{mu}phipartial_{mu}phi-frac{1}{2}m^{2}phi^{2}tag{9.8}$$

$${cal L}_{1}=-frac{1}{2}(Z_{phi}-1)partial^{mu}phipartial_{mu}phi-frac{1}{2}(Z_{m}-1)m^{2}phi^{2}+frac{1}{6}Z_{g}gphi^{3}+Yphi.tag{9.9}$$

Then
$$begin{align}Z_{J}
=~&e^{iint d^{4}x, {cal L}_{1}left[frac{1}{i}frac{delta}{delta J(x)}right]}int {cal D}phi, e^{i int d^{4}x, ({cal L}_{0}+Jphi)}cr
propto~& e^{iint d^{4}x, {cal L}_{1}left[frac{1}{i}frac{delta}{delta J(x)}right]}Z_{0}(J)end{align}tag{9.6}$$

where $Z_{0}(J)$ is (9.5) in the free field theory.

1. Here is my first question: Why is the equal sign in (9.6) is replaced $propto$? In my understanding, since $phi$ is already renormalized, it should be the same as the field operator in free-field theory; thus it should still be $=$ rather than $propto$ .

2. The second problem is I cannot find the part in which the writer proves $Z_{i}=1+O(g^{2})$, nor could I understand why this is required.

By assuming $Z_{g}=1+O(g^{2})$ and taking $Z_{g}=1$, the writer if without $$-frac{1}{2}(Z_{phi}-1)partial^{mu}phipartial_{mu}phi-frac{1}{2}(Z_{m}-1)m^{2}phi^{2}+Yphi,$$
$$langle 0|phi(x)|0rangle =frac{1}{2}igint d^{4}y, left[frac{1}{i}Delta(x-y)frac{1}{i}Delta(y-y)right]+O(g^{3})tag{9.18}$$
where $Delta(x-y)$ is the Feynman propagator.

Clearly (9.18) violates our assumption that $langle 0|phi(x)|0rangle=0$, so $Yphi$ must be incorporated and $Y=O(g)$, but I cannot find the contents describing $Z_{i}=1+O(g^{2})$.

Could anyone help me with these two questions?