Water changes do not hurt the cycle !!!!! This is mis-information any time it is stated. Ammonia and nitrite will injure your fish. Keep the ammonia and nitrite levels safe for your fish. The myth that water changes hurt the cycle is created by folks who do not understand how the bacteria lives reproduces and grows. There is plenty of assumed and mis-guided information in print all over this hobby. If you have doubt about what is and isn't true, research the subject at hand, seperate actual science form random opinions with no information to back them.
Ph should not be a concern at all. You really need a Kh of 2-3 degrees to ensure a stable tank, and a Kh of 2-3 gegrees is going to put you in the 7.6 range with atmospheric levels of co2. In other words if your PH is below 7.6 you should worry otherwise it's not an issue. A Kh test would do you far more good than the Ph test will
Messing with Ph did not hurt your cycle or help it. What did you use to lower the Ph? It will probably cause other problems down the road.
Throwing away the carbon could hurt things because it will colonize bacteria as mentioned. Carbon is unnecessary to use as a rule, and does not work for long when it is used. After the first week (give or take 4 days) the carbon was nothing but bio-media. It's ability to chemically filter is depleted that quickly. If it is in the system, it does serve as a decent bio-filtration media though so don't throw it away randomly.
You can rinse your media in tank water, but as said don't rinse it with tap water. The bacteria do not wash loose easily if at all. I wouldn't worry about excessive cleaning but if you feel the need to rinse you can do so.
The Product Cycle does not contain or claim to contain ammonia. It claims to contain bacteria, but repeated reviews have shown that it does not in fact contain the needed bacteria, and it typically adds a lot of pollution without helping anything. the bacteria it contains )if it's still alive when you buy it) will eat ammonia for the short period they remain alive in your tank. After they die they they do nothing but add to pollution levels. The only viable bacteria in a bottle product I know of is Bio-spira, which would help your situation immensly if you found some and used it.
You do not need or want ammonia if you have fish in the tank. Ammonia at anything but trace levels is deadly to fish. it causes permanent gill damage, and even if your fish survive they will be scarred and their life spans will be shortened because of the gill damage they have endured. Ammonia levels at 0.25 like you were doing at the start are as high as I would let things go. Nitrite likewise needs to be kept low. Killing fish unneccessarily is not a good practice so avoid the advice of "losing a few fish" instead of doing water changes.
So, I assume that now I may be flamed for adding these new fish, but I know people that have had tanks forever and never check more than the PH. The Smithsonian book I just got said to only check the PH too. So, instead of playing with the tank too much, I'm going to ride it out, keep an eye on my fish, and see what happens.
I'm not much for flaming folks once they are already in a really bad situation, but if you ride it out you will be killing more fish in the process. You need to do what is best for your fish which is water changes and constant monitoring until the cycle finishes estabilishing itself in the tank. It is odd that it has taken this long to get rolling, but that is the nature of things. Just protect your fish and hopefully it will not take much longer.
As far as folks who check nothing take care of nothing and claim that is O.K. that's their choice, but it is an irresponsible way to do things. It's no different than putting a terrier in a small apartment and never taking it out for excersize. It will live, but not well. Fish will live in all kinds of nasty condistions, but that does not make it right to keep them that way.
I would avoid any book that considers Ph the only necessary test. as said Ph should be the least of your worries at this point. The fact tht they take this stance points to a really poorly researched book. There is plenty of information an Ph, Kh, Gh, ammonia, nitrite, nitrate and so on in the chemistry article. it is well worth reading. The cycle sticky in the newbie Forum is also full of good information, it does take a while to read, and there is a lot of stuff repeated and corrected but the information is there.
Good luck with things
Dave
Dave