What did Krishna say about trusting him?
His plainest promise is 18.66: take refuge in me alone, I will free you from all sins, do not grieve. In 9.22 he says that for those who think of him without wavering, he carries what they lack and protects what they have.
What does "yogakshemam vahamyaham" mean?
It is the last line of 9.22. Yoga here means gaining what you lack and kṣema means keeping what you have, and Krishna says he himself carries both for the devotee whose mind rests on him.
Does trusting Krishna mean I stop trying?
No. In 2.47 he warns against clinging to inaction, and after hearing 18.66 Arjuna picks up his bow again (18.73). The trust Krishna asks for goes along with the work.
Did Krishna say "leave everything to me and I will take care of it"?
Not in those words. The closest verses are 18.66, where Krishna says take refuge in me alone and do not grieve, and 9.22, where he says he carries the needs of those who rest their mind on him.
Which Gita verse helps when you feel no one is on your side?
9.29: Krishna says he is the same to all beings, and those who worship him with love are in him and he in them. In 9.31 he tells Arjuna to declare it boldly: my devotee never perishes.