The cycle is not about the water, it is all about the rocks. More specific, the dead things in and on the rocks. And how they are disposed of, eaten or rot away. And the good bacteria population growing within the rock, digesting the dead critters.
Yes, there are some free floating bacteria in the water. But is nothing compared to the numbers inside porous rocks.
Lava rock can work. But note that new ones can leach more phosphates into the system than wanted. Also they are not as porous as good quality liverock.
Yip you need live rock. Not sure what you mean by lava rock. lava rock I know, I used to use in my tropical tanks but that wont work in marine i think. (They have very sharp edges etc and I think they not porous enough for bacteria)