I made an extra effort to make the lyrics vaguely sing-able,
but translation accuracy took priority.

生きている いくつもの運命を越えて行く
"ikutsumono" = generally, many "things", not people.
This line is vague; it has several possible interpretations, but the space
between "ikiteiru" and "ikutsumono" implies the former is a stand-alone clause.

人は尽きる命の合間に 出会うことに 怯える間はない
Due to Japanese grammar rules these stanzas cannot be translated in order.
The meaning is very compressed, but "tsukiru inochi no aima" refers to the
interval between life that expires and life after it. It can refer to
either reincarnation or the inability to meet future generations beyond
one's finite life span. "Ma" can mean either time or space.
These are probably not ambiguities, but poetry.

木漏れ日が優しくて でもそれは 永遠ではなくて
"dewanakute" means "is not"

溢れ出る悲しみに 地球はもう蒼くなり
There is no implied possessive in this stanza.
The characters "chikyuu" ("Earth") are sung as "hoshi", which can also
mean "star" or "planet". "Earth" is the correct translation.
"Mou aoku nari" means "has already turned blue", but is read
in the manner of a distant observation, so "is now blue" is a
more compact, contextually accurate interpretation.

人は尽きる命に惑わされ 時に人に 辛くあたるよ
These two stanzas string together in several different ways, because
the subject of the second stanza is implied.
"Toki ni" means "incidentally", which is a jarring choice of words.
Gramatically it can be read as "striking time, striking people" but the
meaning would be more jibberish than poetic.
I can only hear it as an abbreviation of "toki ni wa", which means
"sometimes". It's unconventional Japanese, but that interpretation makes
the most sense (where when people are led astray, they will sometimes be
battered by fate), and with the implied subject the "wa" is unnecessary.

君に光を届けたくて 君の笑顔で癒して
"takute" form means "BECAUSE I want to", which adds a sense of
desperation to these lyrics that cannot be translated into English.
They do not lead into the chorus.