Good job on the first four lines.

Quote Originally Posted by Katina View Post
Как устало солнце биться
How tired the sun is of being beaten
"биться" doesn't mean "to be beaten". It is either "to fight" or "to beat" (as in "the heart is beating"). The word has other meanings too, but I don't think any of those would apply here

Quote Originally Posted by Katina View Post
А я и не знала, как люблю дождь,
And even I don’t know, how to love the rain
And I didn't even know how much I loved the rain.

Quote Originally Posted by Katina View Post
И имя серое твоё - дождь.
And your name is grey - rain
I may be hairsplitting, but I'd say "Your grey name is rain"

Quote Originally Posted by Katina View Post
Встреч знать бы по часам,
Видеть по рукам, сколько мне дышать,
Gibberish. These lines are ungrammatical, so we can only guess about their precise meaning. My guess is: If I could only know the time of our meeting and divine by hands (?) now much time is left for me to breath. But I wouldn't bet a penny that this is what the lyrics author had in mind.

Quote Originally Posted by Katina View Post
Так раз и навсегда,
Like the first (time) and forever
So, once and for all...

Я и не знала...
And even I don’t know...
"And I didn't even know" (or "I never knew", whichever you like better)

Взгляд прочь бегущих лиц
You don’t look at others running
"A glance of the faces that are running away" or "turned-away glance of running faces". This is not a good English, I know, but it isn't a good Russian either.