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This is true. Choose any linear subspace $Z$ (over $K$) of dimension $n- d-1$ and disjoint from $X$, and take $\pi : \mathbb{P}_K^n \setminus Z \to \mathbb{P}_K^d$ the corresponding linear projection. Next, consider a linear automorphism $g \in \mathrm{PGL}(d,K)$ such that $g(Z)$ is the standard linear subspace $Z_0$ defined by the vanishing of the last $n-d-1$ coordinates; $g$ of course depends on $X$.

You can use the projection $\pi' := \pi \cdot g : \mathbb{P}_K^n \setminus Z_0 \to \mathbb{P}_K^d$. It is independent of $X$, hence all $P \in \mathbb{P}^n(\bar{K})$ satisfy $H_K(\pi'(P)) \leq C H_K(P) $ with a constant $C$ depending just on $n$. The original projection, $\pi = \pi' \cdot g^{-1}$, thus sends $\{ P \in X(\bar{K}) \mid [K(P):K] \leq D, \, H_K(P) \leq B \}$ at most $\delta:1$ into the set $\{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K(g' \cdot Q) \leq C \cdot B \}$, where $g' \in \mathrm{PGL}(d,K)$ is a certain element (induced by $g^{-1}$). But the last set is in bijection $Q \leftrightarrow g'^{-1} \cdot Q$ with $\{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K( Q) \leq C \cdot B \} $, and therefore it has the same cardinality. Finally, the latter cardinality is $$ \asymp_C \# \{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K( Q) \leq B \}, $$ as follows for instance from a generalization of Northcott's precise asymptotics to points of a bounded degree, as proved in: [D. Masser, G. Vaaler: Counting algebraic numbers with large height II, Trans. Amer. Math. Soc., 2006.]

This is true. Choose any linear subspace $Z$ (over $K$) of dimension $n- d-1$ and disjoint from $X$, and take $\pi : \mathbb{P}_K^n \setminus Z \to \mathbb{P}_K^d$ the corresponding linear projection. Next, consider a linear automorphism $g \in \mathrm{PGL}(d,K)$ such that $g(Z)$ is the standard linear subspace $Z_0$ defined by the vanishing of the last $n-d-1$ coordinates; $g$ of course depends on $X$.

You can use the projection $\pi' := \pi \cdot g : \mathbb{P}_K^n \setminus Z_0 \to \mathbb{P}_K^d$. It is independent of $X$, hence all $P \in \mathbb{P}^n(\bar{K})$ satisfy $H_K(\pi'(P)) \leq C H_K(P) $ with a constant $C$ depending just on $n$. The original projection, $\pi = \pi' \cdot g^{-1}$, thus sends $\{ P \in X(\bar{K}) \mid [K(P):K] \leq D, \, H_K(P) \leq B \}$ at most $\delta:1$ into the set $\{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K(g' \cdot Q) \leq C \cdot B \}$, where $g' \in \mathrm{PGL}(d,K)$ is a certain element (induced by $g^{-1}$). But the last set is in bijection $Q \leftrightarrow g'^{-1} \cdot Q$ with $\{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K( Q) \leq C \cdot B \} $, and therefore it has the same cardinality. Finally, the latter cardinality is $$ \asymp_C \# \{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K( Q) \leq B \}, $$ as follows for instance from a generalization of Northcott's precise asymptotics to points of a bounded degree, as proved in: [D. Masser, J. Vaaler: Counting algebraic numbers with large height II, Trans. Amer. Math. Soc., 2006.]

This is true. Choose any linear subspace $Z$ (over $K$) of dimension $n- d-1$ and disjoint from $X$, and take $\pi : \mathbb{P}_K^n \setminus Z \to \mathbb{P}_K^d$ the corresponding linear projection. Next, consider a linear automorphism $g \in \mathrm{PGL}(d,K)$ such that $g(Z)$ is the standard linear subspace $Z_0$ defined by the vanishing of the last $n-d-1$ coordinates; $g$ of course depends on $X$.

You can use the projection $\pi' := \pi \cdot g : \mathbb{P}_K^n \setminus Z_0 \to \mathbb{P}_K^d$. It is independent of $X$, hence $H_K(\pi'(P)) \leq C H_K(P)$ with a constant $C$ depending just on $n$. The original projection, $\pi = \pi' \cdot g^{-1}$, thus sends $\{ P \in X(\bar{K}) \mid [K(P):K] \leq D, \, H_K(P) \leq B \}$ at most $\delta:1$ into the set $\{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K(g' \cdot Q) \leq C \cdot B \}$, where $g' \in \mathrm{PGL}(d,K)$ is a certain element (induced by $g$). The last set is in bijection with $\{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K( Q) \leq C \cdot B \} $, and the conclusion is then clear from the Masser-Vaaler generalization of Northcott's asymptotic count.

This is true. Choose any linear subspace $Z$ (over $K$) of dimension $n- d-1$ and disjoint from $X$, and take $\pi : \mathbb{P}_K^n \setminus Z \to \mathbb{P}_K^d$ the corresponding linear projection. Next, consider a linear automorphism $g \in \mathrm{PGL}(d,K)$ such that $g(Z)$ is the standard linear subspace $Z_0$ defined by the vanishing of the last $n-d-1$ coordinates; $g$ of course depends on $X$.

You can use the projection $\pi' := \pi \cdot g : \mathbb{P}_K^n \setminus Z_0 \to \mathbb{P}_K^d$. It is independent of $X$, hence all $P \in \mathbb{P}^n(\bar{K})$ satisfy $H_K(\pi'(P)) \leq C H_K(P) $ with a constant $C$ depending just on $n$. The original projection, $\pi = \pi' \cdot g^{-1}$, thus sends $\{ P \in X(\bar{K}) \mid [K(P):K] \leq D, \, H_K(P) \leq B \}$ at most $\delta:1$ into the set $\{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K(g' \cdot Q) \leq C \cdot B \}$, where $g' \in \mathrm{PGL}(d,K)$ is a certain element (induced by $g^{-1}$). But the last set is in bijection $Q \leftrightarrow g'^{-1} \cdot Q$ with $\{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K( Q) \leq C \cdot B \} $, and therefore it has the same cardinality. Finally, the latter cardinality is $$ \asymp_C \# \{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K( Q) \leq B \}, $$ as follows for instance from a generalization of Northcott's precise asymptotics to points of a bounded degree, as proved in: [D. Masser, G. Vaaler: Counting algebraic numbers with large height II, Trans. Amer. Math. Soc., 2006.]

This is true. Choose any linear subspace $Z$ (over $K$) of dimension $n- d-1$ and disjoint from $X$, and take $\pi : \mathbb{P}_K^n \setminus Z \to \mathbb{P}_K^d$ the corresponding linear projection. Next, consider a linear automorphism $g \in \mathrm{PGL}(d,K)$ such that $g(Z)$ is the standard linear subspace $Z_0$ defined by the vanishing of the last $n-d-1$ coordinates; $g$ of course depends on $X$.

You can use the projection $\pi' := \pi \cdot g : \mathbb{P}_K^n \setminus Z_0 \to \mathbb{P}_K^d$. It is independent of $X$, hence $H_K(\pi'(P)) \leq C H_K(P)$ with a constant $C$ depending just on $n$. The original projection, $\pi = \pi' \cdot g^{-1}$, thus sends $\{ P \in \mathbb{P}^d(\bar{K}) \mid [K(P):K] \leq D, \, H_K(P) \leq B \}$ at most $\delta:1$ into the set $\{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K(g' \cdot Q) \leq C \cdot B \}$, where $g' \in \mathrm{PGL}(d,K)$ is a certain element (induced by $g$). You are thus reduced to an exercise:

Lemma. For any fixed $D \in \mathbb{N}$ and $g' \in \mathrm{PGL}(d,K)$, we have the asymptotic $$ \frac{ \# \{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K(g' \cdot Q) \leq B \}}{ \# \{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K( Q) \leq B \} } \to_{B \to \infty} 1. $$ This is clear anyway from the Masser-Vaaler asymptotic count, which is independent of a choice of projective coordinates. The remainder follows as well from Masser-Vaaler.

This is true. Choose any linear subspace $Z$ (over $K$) of dimension $n- d-1$ and disjoint from $X$, and take $\pi : \mathbb{P}_K^n \setminus Z \to \mathbb{P}_K^d$ the corresponding linear projection. Next, consider a linear automorphism $g \in \mathrm{PGL}(d,K)$ such that $g(Z)$ is the standard linear subspace $Z_0$ defined by the vanishing of the last $n-d-1$ coordinates; $g$ of course depends on $X$.

You can use the projection $\pi' := \pi \cdot g : \mathbb{P}_K^n \setminus Z_0 \to \mathbb{P}_K^d$. It is independent of $X$, hence $H_K(\pi'(P)) \leq C H_K(P)$ with a constant $C$ depending just on $n$. The original projection, $\pi = \pi' \cdot g^{-1}$, thus sends $\{ P \in X(\bar{K}) \mid [K(P):K] \leq D, \, H_K(P) \leq B \}$ at most $\delta:1$ into the set $\{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K(g' \cdot Q) \leq C \cdot B \}$, where $g' \in \mathrm{PGL}(d,K)$ is a certain element (induced by $g$). The last set is in bijection with $\{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K( Q) \leq C \cdot B \} $, and the conclusion is then clear from the Masser-Vaaler generalization of Northcott's asymptotic count.