All tanks have algae to some extent. What types and how bad it is depends on a variety of factors.
First is lighting, algae, cyano bacteria, diatoms, and dinoflagellates all need light for photosynthesis.
Second is nutrients (mainly nitrate and phosphate), all tanks have some nutrients, they are needed for corals, but the idea is to have other organisms (whether they be bacteria, corals, phytoplankton, coralline algae, macro algae, etc.) out compete the undesirable algae for nutrients and real estate, some undesirable ‘algae’ like dinoflagellates (which aren’t actually algae) thrive in tanks with low nutrients where there aren’t enough nutrients to support other algae or organisms to outcompete them (essentially they can thrive on less nutrients than other algae, when there are more nutrients those other algae out compete them).
Third is your CuC and other tank inhabitants. On a real reef, algae and algae like organisms would completely out compete every other organism for nutrients and real estate, but on a real reef there are countless higher lifeforms that eat those algae. Snails, microfauna, fish, urchins, sea stars, crabs, etc. all act as lawn mowers to keep the reef clean and keep algae in balance (there has to be some algae to have a balanced ecosystem). So in your tank, you need to have find the right balance of CuC to the algae your tank produces. In the old days, people would recommend one snail and one hermit crab per gallon, which is a little overboard IMO, I’d start with one per every 3 gallons and then get more if need be. If you have room (a 32g is not large enough) a can’t recommend tangs enough, they graze on algae all day long. You can also get urchins, sea hares, conchs, and other more unique algae eaters, but they can be more expensive and less hardy than snails and hermits (you want a variety of snails and usually just one species of hermits).
And lastly, what determines the types of algae you get in your tank depends on the first and second factors, but also on what kind of algae is introduced to the tank (whether live algae or algae spores). Any fish, coral, rock, water, invert, basically anything that came from another tank or the ocean can introduce strains of algae to your tank. Even dry rock can have algae spores on it (I live around 2k miles from the ocean and have had algae show up in vats of fresh salt water that got just a little bit of light. I had a debate about spores of marine species of algae being airborne recently where someone told me what I experienced was impossible, and so I did some research, and it turns out it’s been studied, sea birds get algae spores on their feathers and then fly up into the air where they dry off and the spores are then entered into the trade winds and can travel thousands of miles, which is how you can end up with marine algae in a vat of salt water that’s had nothing introduced to it 2k miles from the ocean.). So anyway, the type of algae you have depends on what was introduced.
And so to answer your question, the way you set up of cycle your tank really doesn’t affect what types of algae you’ll have, its a variety of factors, but whether you go with live or dry rock doesn’t really matter (although live rock is often already colonized by other organisms so, algae can have a harder time taking root compared to dry rock which is open to whatever can colonize it fastest).
