Not stop = nonstop

Catch the silhouettes of the illegible faces = Catch a glance of blurry faces in silhouette.

Let warm tears stream down the eyelashes = Let my warm tears stream down from my eyes.
{In English it is very weird that tears would "fall from eyelashes". If this sounds weird in Russian too, leave it alone. But, if this line makes sense in Russian, it has to change in English. In English, tears fall from eyes, not eyelashes.

Into the water depths = Into the watery depths. {or} Into the water's depth.

Fog on the water. {This is ok, but we normally say} "Mist upon the water."

To take the whole morning of chill with me without residue, at parting.
{This is awkward sounding, but OK. To make it more English, you should say, (remembering English word order... Something like:}
"to take THIS morning chill with me with no bad feelings, as you leave."
{I know it's different, but it's how we would say something like that.}
{If you say "Morning OF chill" it sounds as if it's a "relaxing" (slang: "chill") morning. "Morning chill" (without the OF) means a "cold morning".}

Emblazing
and vibrating to the beat till dawn
Five multi-colored circles,
{Once again English word order}:
"Five multi-colored circles emblazing and vibrating to the beat till dawn."
{It makes no sense backward.}

And dreams trajectory is getting lost in this flitting
{You can always omit Russian "and" at the beginning of a song stanza, if it has no meaning. If it doesn't connect the previous sentence clause. Russian songs have a lot of "and" in them which make no sense in English. In Russian it's just a "fluff word". In English we use "just" for this. It too has no meaning most of the time, but it's added for "fluff". Eliminate Russian "and" unless it really means "and".}
"Where our dreams lead to gets lost in this fleeting moment."
{Sorry, I know it's very different in English. Dreams don't have trajectories, rockets do. AND despite what your dictionary may say "flitting" is not a word.

Is my reflection drawn or real
In the face of the river? = on the surface of the river?

Till the cold gleam of the sunrise = Till the sunrise's cold gleam shines.
{English needs verbs to be added sometimes, that are implied (but not said) in Russian.}

I'll track you down using the forest breath and dreams trajectory.
{Once again... dreams don't have trajectories in English.}
I'll track you down by the forest's breath and with dream's guidance.

Just a few comments to improve your English. Keep up the good work!