complemented $\ell_p$ subspaces in $\ell_p$ sums of spaces

Question

Note: By "subspace" I always mean an infinite-dimensional closed subspace.

Notation.

Let us write $$\oplus_p\ell_q^n:=\left(\bigoplus_{n=1}^\infty\ell_q^n\right)_{\ell_p}\;\;\;\text{ and }\;\;\;\oplus_0\ell_q^n:=\left(\bigoplus_{n=1}^\infty\ell_q^n\right)_{c_0}.$$ If $X$ is a Banach space, we will also write $$\ell_p(X)=\left\{(x_n)_{n=1}^\infty\in X^\mathbb{N}:\sum_{n=1}^\infty\|x_n\|_X^p<\infty\right\},$$ endowed with the norm $$\|(x_n)_{n=1}^\infty\|_{\ell_p(X)}:=\left(\sum_{n=1}^\infty\|x_n\|_X^p\right)^{1/p}.$$ We define $c_0(X)$ similarly.

Question 1a. Let $1\leq p<\infty$ and $1\leq q\leq\infty$. Does every subspace of $\oplus_p\ell_q^n$ admit a complemented subspace isomorphic to $\ell_p$? More precisely, if $Y$ is a susbpace of $\oplus_p\ell_q^n$, does there exist a subspace $Z$ of $Y$ which is isomorphic to $\ell_p$ and complemented in $\oplus_p\ell_q^n$?

Question 1b. Does every subspace of $\oplus_0\ell_q^n$ admit a complemented subspace isomorphic to $c_0$?

Question 2a. Let $1\leq p\neq q<\infty$. Does every subspace of $\ell_p(\ell_q)$ admit a complemented subspace isomorphic to either $\ell_p$ or $\ell_q$?

Question 2b. Does every subspace of $\ell_p(c_0)$ admit a complemented subspace isomorphic to $\ell_p$ or $c_0$?

I believe the answers to these questions are already known, in which case references would be much appreciated. I suspect something even stronger is true, e.g. that every basic sequence admits a complemented basic subsequence equivalent to $\ell_p$ (or $\ell_q$), but all I really need is a complemented subspace.

I asked this is stackexchange because I thought it would be easy and already known, but maybe it is more appropriate for MO.

Thanks in advance!

Answer 1

Answer to Q1b is particularly easy as this space embeds into $c_0$, hence it is saturated with complemented copies of $c_0$.

There is however a unified argument which works for all your spaces of interest and follows essentially from the principle of small perturbations due to Bessaga and Pełczyński. Please see Proposition 2.4 here. Actually, all you need to do is to unwrap the proof of Proposition 2.1 therein.