The bacteria that convert ammonia to nitrite then nitrite to nitrate need some minerals to grow and reproduce, and to do their work.
Optimum conditions for the fishless cycle should be followed pretty closely. This is when you are trying to grow these bacteria as fast as possible.
After the tank is cycled you can change the water to whatever suits the fish. However, the bacteria may not grow and reproduce too well. That is OK as long as nothing goes wrong. If something kills off the bacteria, however, it will be difficult to grow more in that tank, where the conditions are not optimum.
GH: These bacteria are known to use phosphorus. I do not know if they use any of the minerals that we test as GH. I would make sure the GH is at least 3 German degrees of hardness (roughly 60 ppm). Since plants and fish also need Ca and Mg, even after the cycle is done, I would maintain a minimum of 3 degrees, unless I was trying to breed or raise something (plant or animal) that absolutely needed something less. Higher, even much higher GH is just fine during cycling and after.
KH:These bacteria use the carbon in the carbonates. The KH must stay high enough that they are not shorted during the time you are looking for optimum growth, and should not be deprived after. I would aim to keep the KH at least 3 German degrees of hardness (about 60 ppm) and higher is probably better. The original experiments that isolated and identified these bacteria used a much higher KH to grow them. If the KH drops to zero that may not mean there really is not any carbonate or bicarbonate in the system. Some substrates remove the KH from the water. I do not know if the bacteria can use the KH in the substrate, though, and most bacteria is growing in the filter where all the nutrients come in via the water.
Phosphorus: I would add plant fertilizer KH2PO4 during cycling unless your water already has phosphorus. (just a regular dose based on your tank size, no need to overdo it)
pH: These bacteria grow better in higher pH. I do not know if this is just a side effect of keeping the KH high, but best results seem to happen when the pH is in the upper 7s to low 8s.
I have been following a lot of threads, questions from people having problems during cycling. About half the problems are that the pH is in the 6s, especially under 6.5.
Oxygen: These bacteria use a LOT of oxygen, so keep good water circulation going at all times. Clean the filter as needed. Gently slosh the media in used tank water. Do not do a lot of squeezing or wringing it out when you are still cycling the tank. When the bacteria are well established they hang onto the filter media quite well, and you can get more aggressive with the cleaning.
Ammonia: Keep the ammonia under 5 ppm. More is not good for these bacteria. During the first couple of days of the fishless cycle it is OK to bring it up to 5 ppm, but the moment that nitrites show up allow the ammonia to drop to 3 ppm. Otherwise the first group of bacteria will produce too much nitrite.
This was another problem brought up a couple of years ago:
Do not raise the ammonia to 3 ppm more than once a day. Go ahead and check it as often as you want, but adding more and more ammonia several times a day is not good. This is another way that the nitrites can climb too high. Once a day, 3 ppm is the max that any reasonable fish load can produce, so there is no reason to dose more and more. Conversely, I would bet that you could cycle the tank by dosing less ammonia more often. Maybe... bring it up to 1 ppm three times a day. This would be easier on the plants, and is more like the ammonia production from fish. I have not tried this, though.
Nitrite: Keep it under 5 ppm. These bacteria do not like high nitrite. If you are monitoring the ammonia and only dosing to raise it to 3 ppm once a day or less often, then the nitrites should not get higher than 5 ppm. However the bacteria that remove nitrite are slower growing, and you might need to do a water change. If so, go ahead and drop the nitrites as low as you can. The ammonia to nitrite bacteria will provide a lot more nitrites very quickly.
The scientists that worked out the fishless cycle that I keep posting also tried to grow the nitrite removing bacteria first, knowing that these were the slower growing bacteria. Give them a head start by feeding nitrite to the tank, and waiting to dose ammonia until near the end. It sped up the fishless cycle from 3 weeks to about 2.5 weeks, so not a big gain.
This was the other most reported problem during cycling. The nitrite removing bacteria do not grow as fast as the ammonia removing bacteria. By keeping conditions in the optimum range and not over feeding ammonia the nitrite bacteria will grow, but they are slow. No getting around that.
Salt (NaCl): Different species of bacteria grow in water with different salt levels. The small amount of salt that might be added to fresh water to treat for Ich or reduce Brown Blood Disease is still a small amount, and the fresh water bacteria do just fine in it. As the water gets saltier, however, the bacteria are different. If you are setting up a tank that will ultimately be a brackish water tank I would either cycle it as a brackish water tank, or else make the change from fresh to brackish slowly so the bacteria can adapt. A marine tank has quite different bacteria, and should be cycled as a marine tank.
Temperature: Bacteria grow faster in warmer temperatures. But warmer water holds less oxygen. Optimum temperature for cycling seems to be the upper 70s, and make sure there is really good water circulation. These bacteria really slow down in cold water, and die if they are frozen. They can get overheated and die, too. This is important if you move, and take the aquarium with you. Keep the bacteria as cool as you reasonably can, for example in an air conditioned car. This reduces their need for oxygen, so they are more likely to survive the trip in good condition.
Water movement through the filter, especially the bio media:
Too fast water movement can work against you. This is one of the reasons that bio media with lots of holes-within-holes works so well. There is good flow through the bulk of the media, but the small pores where the bacteria grow have slower water movement. Plenty to keep them well supplied with ammonia and oxygen, but slower so these materials can seep into the biofilm where these bacteria are developing.
Specific answers to your questions:
The pH is too low, probably because the KH is too low. Dose baking soda at the rate of 1 teaspoon per 30 gallons to raise the KH by 2 degrees. You can raise it higher, if you want. You could also dose potassium bicarbonate if you think the sodium in baking soda would cause problems.
Coral sand adds minerals, raises GH, KH and pH. Do not ADD coral sand when you want the GH to go DOWN.
You can blend reverse osmosis water with your tap water to help lower the mineral (GH) level. This will also lower the KH level, so dose baking soda or other source of carbonates as needed. Top off with RO, not tap.
When water evaporates it leaves the minerals behind. Constantly topping off with tap water is adding more and more minerals to the tank.
The GH lowering pillows work, but not for large volumes of water. I would rather use RO + Tap until the GH is as high as you want it, then add the right minerals so the KH is at optimum levels.
Here is a way to test the GH, and find out what it is without using all the reagent or overflowing the test tube:
Put 1 ml of aquarium water + 4 ml of RO in the test tube. Each drop of reagent now tells you 5 degrees, so you count:
1 drop = 5 degrees
2 drops = 10 degrees
3 drops = 15 degrees...