I think it has had a detrimentally influenced effect on my speaking skills, although I speak quite consistently, several times per week and mostly with natives.

More than 8 months ago, I've embarked on reading classical literature in English.

Every day I look forward to the evening, whenand after taking a hot shower and writing my daily diary entry, I turn on my e-reader and delve deep into captivating worlds of literature.

When I look up a word, I never forget to check it so that later I could download the list of all previously unknown words later.

I've been doing it for quite a while and have already accumulated more than 500 words learned exclusively via reading.

But of course it was not, forn’t, because coming across these words while reading and not stopping to look them up brings me so much joy!

I think it has been detrimentally influenced to my speaking skills, although I speak quite consistently, several times per week and mostly with natives.

More natural phrasing here, "detrimentally influenced" is a little too verbose to be natural here I think.

More than 8 months ago, I've embarked on reading classical literature in English.

Since you specified "more than 8 months ago", you're talking about a single point in time at the beginning, rather than your current state of being, so it's more natural to say "I embarked" (describing the action of embarking) than "(I have / I've) embarked" which is describing you as being in the state of having embarked. If you didn't specify a specific point in time, then "I've embarked" would be more natural.

(This is pretty nitpicky feedback - both are perfectly understandable)

When I look up a word, I never forget to checmark it so that later I couldan download the list of all previously unknown words.

Tense is a little finicky here, but once you explain why you take the action, you're no longer in a conditional context.

I think it has detrimentally influenced my speaking skills, although I speak quite consistently, several times per week and mostly with natives.

"quite consistently" is correct, but it may not be the best way to convey your meaning because each time you speak, it is different--so it's not like doing the same activity regularly over time. My first thought is "frequently." But that's explained in "several times a week" so maybe the best revision is to shorten the sentence to sharpen the focus: "although I speak several times a week with native English speakers."

More than 8 months ago, I've embarked on [ ] reading classical literature in English.

Replace "embarked" with "started" in your mind: you only embarked or started once. However, "embarked" needs more than "started" to feel complete. For example, you could say "embarked on a journey, an adventure, a course of reading" Because it literally refers to getting on a boat, plane, (not sure about a train), it needs supporting words.

Every day I look forward to the evening, when after taking a hot shower and writing my daily diary entry, I turn on my e-reader and delve deep into captivating worlds of literature.

There's nothing wrong with the plural, but it tends to be used as a singular noun in the expression " the captivating world of literature." (Within each book you may enter a different world, but you are probably delving into one book world at a time; and they are all part of classical English lit.)

When I look up a word, I never forget to check it so that later I couldan download the list of all previously unknown words.

Since you actually to check them, I think "can" is right. It also sounds right grammatically.

In fact, they are being so seldom used that sometimes I think that it was all in vain.

"being used so seldom" is confusing.
If you mean that YOU are using them seldomly, then make the sentence active: "I use them so seldomly" or "rarely"
If you mean used by everyone, or everywhere, or in conversation, etc. then that needs to be clarified.