1) 이게 is a contraction of 이것 +이(suffix) So 이게=이것이
2) 이거 is a contraction of 이것. So 이거=이것
3) 이것,거 is 이(this) + 것,거(thing). So 이것 means 'this thing.'
이게 and 이거 both mean 'this thing' but they are different grammatically.
While you can say '이거(-를) 먹어도 되나? as in, can I eat this(-thing)?' you can't say 이게(이것이) 먹어도 되나?