Hello.
Disclaimer. I am not a linguist, I have technical background and do not speak English myself (I do not do it as a part of my regular everyday activity).
Some people, especially Slavic ones, do not make the difference in the pronunciation of the words, that differs by the long e and short i sounds, like beach-bitch, peace-piss, sheet-shit, beam-bim. Luckily, we pronounce all of them with long e sound. I decided to teach myself to pronounce it right and found an interesting (at least for me) phenomenon. We are taught that English does not have soft consonants the way Russian does. I believe that soft consonants we make the following way – we smile a little and raise the middle of the tongue in the upper position as for the long e sound: b b’. Our textbooks claim that the difference in “beat” and “bit” is in the vowel sound only. But if you start pronouncing the hard b in “beat” and then raise the middle of your tongue up for the long e sound for a very short time it will sound as the short i. But if you raise your tongue in advance and start pronouncing b in this position you will get traditional Russian soft b’ sound.
So, I wonder if it is valid to pronounce “beat” with the Russian soft b’ sound, i.e. if you raise your tongue in advance for long e sound and start pronouncing b in this position. If you start with the soft b’ it will be possible to continue with the long e only. But how it will sound for native English speaker: as a native accent, a foreign accent or just normal? And if you find this variant of pronunciation acceptable it will mean (from my point of view) that English does have soft consonants too and you just emphasize the vowels after them.

One problem is assimilation, where any two adjacent sounds will at least subtly, or sometimes obviously, affect each other's pronunciation. This makes describing and understanding Russian pronunciation in technical terms difficult, because there are differences in how the consonants are pronounced, but also the vowels. In fact, while it is generally described as a difference in the consonants, it is actually written in the language mostly via the vowels!

To fully understand this, you should read about palatalization (for example, on Wikipedia), including a bit of the history of Russian.

Generally speaking, palatalization is not a general part of English pronunciation, so most consonants are just not palatalized, nor are the vowels that follow. But we do have some similarities. A few consonants pronounced near the palate do get pronounce that way and therefore change the vowels near them, or sometimes palatal (high front) vowels cause consonants to palatalize via assimilation. But this isn't a relevant factor for example with [b]. There are also some interesting details like with how <u> is often pronounces "yu", like in "union".

The "short/long" distinction of English vowels is generally unrelated to this, but also tricky, and as you say, not found in Russian. Speakers of many languages have trouble with this (for example, Spanish), so it's familiar to English listeners, who can probably still understand in context. (Very rarely does it actually matter whether you say "sheet" or "shit", because those are such different meanings it's clear anyway. And yes, it's helpful that you probably would make an error by pronouncing the longer for I guess!) But the "short/long" distinction is tricky because of English spelling: these aren't actually the same sounds, just shorter/longer. They're different qualities of vowels. English spelling was standardized about 500 years ago, and around that same time the Great Vowel Shift was affecting the pronunciation of long vowels, but not short vowels. So the short vowels mostly remained in their original positions (you can see this by comparing their pronunciation to Latin or various other languages, or even transliterated Russian), but the long vowels are now just different sounds, about "one place" away within the vowel space. But then we have pairs like "ee" [i] vs "i" [ɪ], which you identified, but which are not generally called "short/long" pairs by English speakers, although linguists know this of course.

The example with [i] is a little unusual, because even in Russian, и vs. ы is clearly a vowel difference, rather than (only) a consonant difference, which is distinct from the other vowels where there is a clear "y" [j] sound which we could either consider as part of the consonant (palatalized) or as a diphthong for the vowel. So for example Russian я is basically a combination of йа, rather than a distinct vowel.

Anyway, for English, there is no such thing as a soft consonant. That's just not part of the grammar (phonology). There are a few consonants that are always pronounced like "soft" Russian consonants (they're palatalized), but there is no variation in this way. The front vowels [i, e] also do create to some extent a little palatalization, but that's not nearly as extreme or important as in Russian. On the one hand, because this is not an important (phonemic) contrast in English, you can in principle pronounce them either way without changing the word to a different meaning. On the other hand, (most) English consonants just aren't "soft" (they're "hard" if you need a label), so English speakers would pronounce them "normally" (for English), and any variation from that would sound like a "foreign" accent (but probably still understandable because there's no other word with "soft" variants that would be confused). There are also a few cases where coincidentally the same kind of pronunciation as in Russian might appear, as in beauty (roughly бюти [bjuti]), but not because English has a soft/hard contrast, just because similar sounds go together in some cases. As for [bi] specifically (or any consonant followed by [i]), there is just never a contrast, and we pronounce the vowel like и, not ы.

(As an interesting footnote, English speakers do pronounce ы in the very specific context of some quickly-pronounced words [mostly function words or simple adverbs?] like "just" as in "I just did it", but not when the word is stressed. Whereas most vowels reduce to schwa [ə], in just a few words instead the vowel reduces to a similar but raised form [ɨ], although it's really more of a lax version of that, which I guess we could write as [ɪ̵].)

2. I had never met a Russian who would thought of short i as about Russian ы sound.

I was comparing a few things above. That's not exactly the same sound, although it's also not so different. (ы is in the center of i/u, while long or short i is in the front.)

Ы is the American ash sound (as in bad) with the jaw put forward and a very little bit up. You can check it yourself, just say the ash and move your jaw back and forth.

Most vowels can be described in similar ways, just moving around the vowel space. Those two aren't especially close. /æ/ is a low front vowel,* while ы is a high central vowel. It doesn't really have a clear English equivalent, although it's sort of a mix between i/u. It's farther back than /i/, farther forward than /u/, and unrounded like /i/.
[*One interesting detail is that although /æ/ is front, it actually is not articulated as far forward as /i/ due to the shape of the mouth, which isn't a perfect square: it's wider for high vowels than for low vowels, which instead extend barely past central vowels, so in a sense you could say that ы is almost above /æ/, but not generally in terms of how phonological contrasts work, but just coincidentally due to the shape of the mouth.]

I'd recommend you do some reading about phonetics if this interests you. The very short version is that for vowels, we can describe them based on three features: backness, height and rounding. And for consonants, we can similar describe them via place, manner and voicing. (There are a few more details in specific languages, like palatalization in Russian, but those three features for vowels and consonants cover the basics for all languages.) Take a look at Wikipedia (or other website of your choice), and try to understand the International Phonetic Alphabet's organization in this way. (Note that the goal isn't memorizing the symbols, but understanding how one relates to another. Like b is a voiced version of p, or i is a higher version of e, etc.)

In my opinion, the differences are too big to consider these phonemes as allophones.

I didn't claim they are allophones. I'd assume they're contrastive (phonemic), except that you instead might want to claim that the consonants all come in soft/hard pairs. So the problem is that you only need one contrast: consonants or vowels. You don't need to have both, because they predict each other. This problem gets complicated because it requires a specific analysis identifying the locus of the contrast, which is not easily determined from the actual data. A similar example from English is that before voiced consonants, vowels are pronounced longer than the same vowel before a voiceless consonant: compare "bad" (longer æ) and "bat" (shorter æ). Almost no one would claim that vowel length in those cases is phonemic, yet it is a perceptually salient clue to the identify of the word, because the final stops are often unreleased so it's hard to hear any difference, and we actually do rely on the vowel to figure out what the consonant is. The situation for Russian seems similar to me.

So, beat, bit and быт (everyday life) are three different phonations for me.

I agree.

In the end, there's no substitute for experience: listen, and then practice pronouncing. Repeat (for years). Even for linguists, identifying these details is only a small part of the problem, and the rest is just practice. And once you get to a very narrow level of detail, you'll find native English speakers vary too (at least from different regions, and sometimes even just different individuals). A range of pronunciations is acceptable, although the details, and assimilation (the way that two adjacent sounds change each other) is crucial to really sounding convincingly native (if that's your goal). Being understood is a lot easier, of course. And just remember: all that native speakers can judge your accent by is just their experience (a lot of experience), which means it won't follow rules, just tendencies, and sometimes very subtle ones. If you don't believe that, then just look at how easily you can recognize someone speaking with your own accent (your own home town, etc.) if you're traveling. We all have accents, and our ears are very good at noticing them, down to the smallest details.