Foam can sometimes add to nitrate headaches as they fill with particulate matter and create a fabulous place for bacteria to grow. They really need to be cleaned every 3-4 days and at the very most weekly. If you're really needing to keep the water free of particulates you can still keep the foam but need to be more diligent about keeping it clean. On the flip side, removing the foam and replacing with carbon will also help for great water polishing.
The ceramic just provides a substrate for bacteria to cling to much in the way bioballs or your live rock works. Seeing as how you have so much live rock in the tank I would probably remove them as they could perhaps be contributing to nitrates as well, and really you dont' need them.
What is your water source? Do you use RO/DI (reverse osmosis, deionized)? It could be that if you are using tap water you are actually adding phosphates to your tank as you do water changes and that could be a source to look into. If you are really concerned about it and want to continue to use tap water, try phosban or another type of phosphate remover. You can use this in your canister filter using a small filter bag.
All you really need to clean on your skimmer is the cup, sometimes the neck does get a bit of buildup and you can just reach in and swipe at it with a paper towel. Every 3-6 months (depending on how motivated I am) I remove mine and give it a thorough cleaning with vinegar and water just to remove any crud thats building up in there. A biofilm can develop on the walls of your skimmer decreasing its performance so its good to get rid of it from time to time.
You can certainly remove your sandbed if you don't want it, I run a barebottom tank myself and while I do miss the look of sand I think my tank (and myself) have benefited more without it than with. You can either remove it bits at a time (say 25% of it just prior to a water change so you can suck out any nasties that may have been stirred up). Or totally go to town on it depending on how labor intensive you want your day to be
Hope this helps
