Embeddings of forcing notions - preserve properness? Let $ M $ be a countable, transitive model for $ \mathsf{ZFC}^* $, i.e. for a sufficiently large finite fragment of $ \mathsf{ZFC} $. Suppose that $ \mathbb{P} := (P, {\leq_P}, \mathbb{1}_P) \in M $ and $ \mathbb{Q} := (Q, {\leq_Q}, \mathbb{1}_Q) \in M $ are forcing notions.

Reminder
Definition ([Kun80, VII.7.1]). A mapping $ i \colon P \to Q $ is a complete embedding of $ \mathbb{P} $ into $ \mathbb{Q} $ iff
(i) $ \forall r, s \in P \ (r \leq_P s \implies i(r) \leq_Q i(s)) $,
(ii) $ \forall r, s \in P \ (r \perp_P s \implies i(r) \perp_Q i(s)) $, and
(iii) $ \forall q \in Q \ \exists p \in P \ \forall r \in P \ (r \leq_P p \implies i(r) \parallel_Q q) $.
A condition $ p \in P $ as in (iii) is called a reduction of $ q $ to $ \mathbb{P} $.
Remark. Note that in clause (ii) equivalence holds because of (i).
Definition ([Kun80, VII.7.7]). A mapping $ i \colon P \to Q $ is a dense embedding of $ \mathbb{P} $ into $ \mathbb{Q} $ iff
(i) $ \forall r, s \in P \ (r \leq_P s \implies i(r) \leq_Q i(s)) $,
(ii) $ \forall r, s \in P \ (r \perp_P s \implies i(r) \perp_Q i(s)) $, and
(iii) $ i[P] $ is dense in $ \mathbb{Q} $.
Remark. Every dense embedding is a complete embedding.
Theorem ([Kun80, II.3.3]). There exists a dense embedding of $ \mathbb{P} $ into $ \operatorname{RO}(\mathbb{P}) \setminus \{ \mathbb{0} \} $.
Lemma ([Kun80, VII.Ex.C2]). If $ \mathbb{P} $ and $ \mathbb{Q} $ are separative and $ i \colon P \to Q $ is a complete embedding of $ \mathbb{P} $ into $ \mathbb{Q} $, then $ i $ is one-to-one, $ i(\mathbb{1}_P) = \mathbb{1}_Q $, and $ (r \leq_P s \iff i(r) \leq_Q i(s)) $ holds for all $ r, s \in P $.

Now, consider the following statements:
(C1) For each $ \mathbb{Q} $-generic $ H $, there exists a $ G \in M[H] $ such that $ G $ is $ \mathbb{P} $-generic over $ M $. (Then $ M[G] \subseteq M[H] $.)
(C2) For each $ \mathbb{P} $-generic $ G $, there exists a $ \mathbb{Q} $-generic $ H $ such that $ G \in M[H] $. (Then $ M[G] \subseteq M[H] $.)
(C3) There exists a complete embedding $ i \in M $ of $ \mathbb{P} $ into $ \mathbb{Q} $.
(C4) There exists a complete embedding $ i \in M $ of $ \mathbb{P} $ into $ \operatorname{RO}(\mathbb{Q}) \setminus \{ \mathbb{0} \} $.
(D1) For each $ \mathbb{Q} $-generic $ H $, there exists a $ G \in M[H] $ such that $ G $ is $ \mathbb{P} $-generic over $ M $ and $ H \in M[G] $. (Then $ M[G] = M[H] $.)
(D2) For each $ \mathbb{P} $-generic $ G $, there exists a $ \mathbb{Q} $-generic $ H $ such that $ G \in M[H] $ and $ H \in M[G] $. (Then $ M[G] = M[H] $.)
(D3) There exists a dense embedding $ i \in M $ of $ \mathbb{P} $ into $ \mathbb{Q} $.
(D4) There exists a dense embedding $ i \in M $ of $ \mathbb{P} $ into $ \operatorname{RO}(\mathbb{Q}) \setminus \{ \mathbb{0} \} $.
(Pr) If $ \mathbb{Q} $ is proper, then $ \mathbb{P} $ is also proper.

Question
What implications between the above statements are provable in $ \mathsf{ZFC} $? Which are not?
If it is helpful, you may assume that $ \mathbb{P} $ and $ \mathbb{Q} $ are separative partial orders (in the strict sense).

Main problem
Suppose that (C1) holds. What additional assumptions do we need to show (Pr)?
(Note that (C2) implies (Pr). So what additional assumptions does one need to show (C2) from (C1)?)

Bibliography
[Kun80] Kenneth Kunen: Set Theory: An Introduction to Independence Proofs. North Holland, 1980
 A: The following implications are either trivial or well known:
(D$ n $) implies (C$ n $): Trivial.
(C3) implies (C1): Use $ G := i^{-1}[H] $. (See [Kun80, VII.7.5].)
(C3) implies (C4): The composition of two complete embeddings is a complete embedding.
(D3) implies (D1): Use $ G := i^{-1}[H] $. Then $ H = \{ q \in Q : \exists p \in G \ i(p) \leq q \} $. (See [Kun80, VII.7.11].)
(D3) implies (D2): Use $ H := \{ q \in Q : \exists p \in G \ i(p) \leq q \} $. Then $ G = i^{-1}[H] $. (See [Kun80, VII.7.11].)
(D3) implies (D4): The composition of two dense embeddings is a dense embedding.

Regarding (C1) does not imply (C3) and (C1) does not imply (C4):
Let $ \mathbb{Q} $ be the Cohen forcing and let $ \mathbb{P} $ be the lottery sum of $ \mathbb{Q} $ and $ \operatorname{Col}(\omega, \omega_1) $. Then (C1) clearly holds (use $ G := H $), but (C3) and (C4) do not hold since $ \mathbb{P} $ is not c.c.c. whereas $ \mathbb{Q} $ and hence $ \operatorname{RO}(\mathbb{Q}) $ satisfy the countable chain condition.
(See the comments of Yair Hayut and Andreas Blass.)
Also see A common forcing extension obtained via different forcing notions by J.D. Hamkins or Lemma 25.5 in T. Jech's book Set Theory (1978 edition).

Regarding (C3) implies (C2):
Each $ H \subseteq Q $ which is $ \mathbb{Q} / G $-generic over $ M[G] $ is also $ \mathbb{Q} $-generic over $ M $, and
$$
G \in M[G][H]_{\mathbb{Q} / G} = M[H]_{\mathbb{Q}}.
$$
(See [Kun80, VII.Ex.D3] and [Kun80, VII.Ex.D4].)
