I am not really sure what you mean by unestablished clean live rock.
No it's not necessary to buy the live rock from the store. People cycle using no live rock and if you do have live rock you are ahead of the game. If you want coraline you should introduce some to the tank. You can even buy 1/100 of a pound of rocks with coraline on it to start.
Sorry to tell you but you have nice dead rock. And yes you can cycle without live rock or coralline covered rock. Read about the different ways to cycle, pick one way and follow it through. After that you can decide whether to add some live rock which will give you some critters or you can just buy the critters you want. And if you like coralline you can add one small caraline covered rock and over time it should spread.
I would wait a month after the cycle completes. Critters need what to eat. Start with adding copods and amphipods and a couple small bristle worms. You can probably get them from MR members. Copods are the hard ones to get but sometimes it is sold here. Or you can get some chaeto from here and hope there are pods in it. Eventually you will have them whether you buy them or not because they will hitchhike on stuff you buy.
You can speed up everything by adding some fully established rocks form an old system and it helps getting the critters going. As for coraline growth you can get a piece of rock that is covered and crub it with a toothbrush and let the particles fall to your existing rocks. I did that with my 55 and it looks fully matured in 5 months.
If you want a coralline rock take a rock with coralline scrape it of and sprinkle it around your tank. If you want quick growth add calcium. I would also collect algae from different people. Not always but usually the bad stuff is on rock not algae. Algae is easy to scan for things.