1. The linear sigma model (11.14) has 3 terms, and hence 3 counterterms, and hence needs 3 renormalization conditions (11.17a+b+c).

2. Yes, since the $N$th component $$\phi^N(x)~=~ v +\sigma(x) \tag{11.8}$$ and the VEV$^1$ $$ v~=~\frac{\mu}{\sqrt{\lambda}},\tag{11.7}$$ then condition (11.16) for a connected 1-point function is equal to $$\langle \sigma \rangle^c_{J=0}~=~0,$$ i.e. that the tadpoles for the $\sigma$ field vanish$^2$ $$ \left(\fbox{1PI}==\stackrel{\sigma}{=}==\right)_{\text{amputated}}~=~0,\tag{11.17a}$$ cf. e.g. my Phys.SE answer here.

3. Yes, diagrams with more and more interaction terms do enter on the LHS of eq. (11.17a). For explicit calculations at one-loop, see eqs. (11.31) & (11.32).

4. Yes, the VEV and the coupling constants do run, but the description in terms of the Lagrangian (11.14) and the Mexican hat potential remains valid despite the RG flow.

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$^1$ Here the mass parameter $\mu$ in the Lagrangian (11.14) is the renormalized mass. It should not to be conflated with the physical mass $m$, which satisfies
$$ \left(==\stackrel{\sigma}{=}==\fbox{1PI}==\stackrel{\sigma}{=}==\right)_{\text{amputated}}~=~m^2-\mu^2\quad \text{at}\quad p^2=m^2.$$

$^2$ The tadpoles for the $\pi^k$ fields vanish automatically, due to a $\mathbb{Z}_2$ symmetry.

That's a good question.

1. The linear sigma model (11.14) has 3 terms, and hence 3 counterterms, and hence needs 3 renormalization conditions (11.17a+b+c).

2. Yes, since the $N$th component $$\phi^N(x)~=~ v +\sigma(x) \tag{11.8}$$ and the definition of the VEV$^1$ $$ v~:=~\frac{\mu}{\sqrt{\lambda}},\tag{11.7}$$ then condition (11.16) for a connected 1-point function is equal to $$\langle \sigma \rangle^c_{J=0}~=~0,$$ i.e. that the tadpoles for the $\sigma$ field vanish$^2$ $$ \left(\fbox{1PI}==\stackrel{\sigma}{=}==\right)_{\text{amputated}}~=~0,\tag{11.17a}$$ cf. e.g. my Phys.SE answer here.

3. Yes, diagrams with more and more interaction terms do enter on the LHS of eq. (11.17a). For explicit calculations at one-loop, see eqs. (11.31) & (11.32).

4. It is important to realize that the field $\phi~=~\phi_0/\sqrt{Z_{\phi}}$ and the VEV (11.7) scale differently under the RG flow. In particular, since both sides of eq. (11.7) should remain$^3$ finite/physical, one should not try to e.g. introduce infinite $Z$-factors and/or counterterms into eq. (11.7), cf. Refs. 1 & 2.

References:

1. M.E. Peskin & D.V. Schroeder, An Intro to QFT, 1995; p. 353-355.

2. M. Srednicki, QFT, 2007; chapter 31. A prepublication draft PDF file is available here.

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$^1$ Here the mass parameter $\mu$ in the Lagrangian (11.14) is the renormalized mass. It should not to be conflated with the physical mass $m$, which satisfies
$$ \left(==\stackrel{\sigma}{=}==\fbox{1PI}==\stackrel{\sigma}{=}==\right)_{\text{amputated}}~=~m^2-\mu^2\quad \text{at}\quad p^2=m^2.$$

$^2$ The tadpoles for the $\pi^k$ fields vanish automatically, due to a $\mathbb{Z}_2$ symmetry.

$^3$ Later in dimensional regularization one typically makes the coupling constant $\lambda\to\lambda\tilde{\mu}^{\epsilon}$ dimensionless, cf. Ref. 2.

1. The linear sigma model (11.14) has 3 terms, and hence 3 counterterms, and hence needs 3 renormalization conditions (11.17a+b+c).

2. Yes, since the $N$th component $$\phi^N(x)~=~ v +\sigma(x) \tag{11.8}$$ and the VEV$^1$ $$ v~=~\frac{\mu}{\sqrt{\lambda}},\tag{11.7}$$ then condition (11.16) for a connected 1-point function is equal to $$\langle \sigma \rangle^c_{J=0}~=~0,$$ i.e. that the tadpoles for the $\sigma$ field vanish$^2$ $$ \underbrace{\fbox{1PI}==\stackrel{\sigma}{=}==}_{\text{amputated}}\qquad =~0,\tag{11.17a}$$ cf. e.g. my Phys.SE answer here.

3. Yes, diagrams with more and more interaction terms do enter on the LHS of eq. (11.17a). For explicit calculations at one-loop, see eqs. (11.31) & (11.32).

4. Yes, the VEV and the coupling constants do run, but the description in terms of the Lagrangian (11.14) and the Mexican hat potential remains valid despite the RG flow.

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$^1$ Here $\mu$ is the renormalized mass (not to be conflated with the physical mass $m$).

$^2$ The tadpoles for the $\pi^k$ fields vanish automatically, due to a $\mathbb{Z}_2$ symmetry.

1. The linear sigma model (11.14) has 3 terms, and hence 3 counterterms, and hence needs 3 renormalization conditions (11.17a+b+c).

2. Yes, since the $N$th component $$\phi^N(x)~=~ v +\sigma(x) \tag{11.8}$$ and the VEV$^1$ $$ v~=~\frac{\mu}{\sqrt{\lambda}},\tag{11.7}$$ then condition (11.16) for a connected 1-point function is equal to $$\langle \sigma \rangle^c_{J=0}~=~0,$$ i.e. that the tadpoles for the $\sigma$ field vanish$^2$ $$ \left(\fbox{1PI}==\stackrel{\sigma}{=}==\right)_{\text{amputated}}~=~0,\tag{11.17a}$$ cf. e.g. my Phys.SE answer here.

3. Yes, diagrams with more and more interaction terms do enter on the LHS of eq. (11.17a). For explicit calculations at one-loop, see eqs. (11.31) & (11.32).

4. Yes, the VEV and the coupling constants do run, but the description in terms of the Lagrangian (11.14) and the Mexican hat potential remains valid despite the RG flow.

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$^1$ Here the mass parameter $\mu$ in the Lagrangian (11.14) is the renormalized mass. It should not to be conflated with the physical mass $m$, which satisfies
$$ \left(==\stackrel{\sigma}{=}==\fbox{1PI}==\stackrel{\sigma}{=}==\right)_{\text{amputated}}~=~m^2-\mu^2\quad \text{at}\quad p^2=m^2.$$

$^2$ The tadpoles for the $\pi^k$ fields vanish automatically, due to a $\mathbb{Z}_2$ symmetry.