Which properties of $\prod_i E_i$ endowed with $(x,y)↦\sup_id_i(x_i,y_i)$ or $(x,y)↦\sum_i2^{-i}\min(1,d_i(x_i,y_i))$ can we infer?

Question

I know there are similar questions on the board, but they don't answer all of my questions: Let $I$ be a countable set, $(E_i,d_i)$ be a metric space for $i\in I$ and $E:=\prod_{i\in I}E_i$.

Can we define a metric on $E$ such that

1. $E$ is complete, if each $E_i$ is complete?
2. $E$ is compact, if each $E_i$ is compact?
3. the induced topology is the product topology?

There are two candiates which come to my mind: $$d_1(x,y):=\sup_{i\in I}\rho_i(x_i,y_i)\;\;\;\text{for }x,y\in E$$ and $$d_2(x,y):=\sum_{i\in I}a_i\rho_i(x_i,y_i)\;\;\;\text{for }x,y\in E,$$ where $(a_i)_{i\in I}\subseteq(0,\infty)$ with $\sum_{i\in I}a_i<\infty$ and $\rho_i$ is a metric on $E_i$ bounded by $1$ and generating the same topology as $d_i$ (for example, $\rho_i:=\min(1,d_i)$ or $\rho_i:=d_i/(1+d_i)$; it shouldn't matter which one we pick) for all $i\in I$.

Which of 1. to 3. do $d_1$ and $d_2$ satisfy?

From the other questions I only know that $d_2$ satisfies 1. and 3.

Answer 1

If we choose $d_2$ then 3. holds as I show here, in essence.

Fact 2. then follows by Tychonoff's product theorem and 1. is quite standard to see, and well-known.

With $d_1$, 3. will not hold but we get a strictly finer topology than the product one, and 2. fails, also for connectedness. I do believe that 1. still holds, the completeness of $\ell_\infty$ is a special case.