借给我自行车???? means lend your bike to me. 借我自行车has two meaning: the first is lending your bike to me, the whole sentence should be 借给我自行车,but we usually say 把你的自行车借给我. The second is you borrow my bike, the whole sentence should be 借我的自行车。

You're abusolutely right. 给 was omitted.

但感觉怪怪的。it's a little bit weird though. 我们一般说usually we say: 把你（的）自行车借我（骑骑） lend me your bike.