I see several issues that should be addressed.
The nitrates are way too high. To get them down is going to require a lot of partial water changes. Typically you want to change 1/3 to 1/2 the water at a time, and do about 3 or 4 of them about a week apart to get the nitrates and phosphates down. A 50% water change means a 50% reduction in nitrates and phosphates. Be sure to carefully match SG and temp when doing these big water changes.
Live rock that get coated in algae like that may have absorbed a certain amount of nitrate and phosphate. If you make the water change and the level is down, but then goes way back up in a few days, you make have this issue.
Get rid of the rubble rock in the filtration system. This tends to become a massive nitrate factory and oxygen consumer. If you want to rinse off the rubble rock in SW to get it clean, and them put it in the main display tank ok. It's ok there because you can get to that to clean it. You can keep the calupera in there as long as it's not becoming a mechanical filter.
I highly recommend a skimmer. It's one of the few devices that can remove waste products from the water before the biological filtration needs to break it down. Expect to spend some money on this, and avoid the junk low end skimmers. I used an AquaC Remora on my 30 gal reef, but there are a lot of other good products out there today.
Consider using GFO in the filtration system. The most effective way to do this is in a reactor, but it can be placed in the regular filtration system. You can buy Phosban, but it's much less expensive to get it in bulk from an online source.
When doing the water changes, test your new water. It's possible that you have something wrong with your RO/DI unit. New water should read 0 or close to it.
Many people tend to over feed the tank, but I don't think you have this problem. Two things to consider. Dry foods tend to contain a lot of nitrates and phosphates. Frozen foods should be thawed and rinsed off in a net first. Don't just drop in a cube.
Your mix of snails is about right. You might want to add more to take care of the algae. You can easily as high as 1 to 2 per gal of water, depending on their size. You can ramp this up too. Add 10 or so, and see how much they help. If needed add another 10.
You have a wrasse in your tank, so you don't want to reduce the sandbed. They need a place to burro. Also, in looking at your tank picture, you don't seem to have a lot of algae growing on it, so I don't think this is a big factor. You may want to vacuum the sandbed. Only go down about 3/4" or so to get out the worst dirt. You can often just do the open areas. This is a good thing to do while your making those large water change mentioned above.
How long has it been since you changed your light bulbs? If it's been more that 18 months, replace them. Of course, you could also upgrade the lighting, but at this point I think your best goal is to get the algae issues fixed first. It is possible to cut back on the number of hours your lighting the tank, but I'd not go below 8. Since you corals are doing well, go carefully here. Make any lighting change slowly.
