OK, let's see what we can do to help. First thing, it is quite common to have issues vastly different from tank to tank. What may have "worked" once may not work a second time due to varying water parameters, different fish (Some fish are "hardier" than others, or physiologically they function differently), substrate, if there are plants or not, etc...so don't let that bother you. But there are some guidelines to ensure success.
An aquarium is an unique closed system. The fish depend upon the water for life as we depend upon air; but our air is (for purposes of this discussion) limitless--even in our homes, there is "fresh" air constantly coming in, unless you manage to have a completely sealed room. This is quite the opposite in the aquarium; the water is finite, and the fish cannot escape from that water no matter what may occur chemically to the water's physical properties. In nature, the fish can move elsewhere in the stream or lake if the temperature falls or rises, of something toxic enters the water; not in the aquarium, where the ration of fish to water is vastly higher than it would ever be in most natural systems. This is important to understand for all aquarists, because when we cause a change to the water it can have drastic consequences. And changes to the water usually affect the physiology of the fish. As one small example, when the pH of the water changes, the fish must change the pH of its blood to match, because water is constantly passing through the cells of the fish--we call this osmosis--and the fish has to regulate its pH. Same with temperature, and so on. This is why sudden fluctuations in these things can be dire for the fish.
I suggest using your tap water for the reason you give; a pH difference of .2 is not worth the fuss. And use a good conditioner [more on which below]. A comment on no conditioner with bottled water--that may or may not be sensible, depending upon what is in the bottled water. I won't get into all that as you're going with tap now. And as for the chlorine previously, yes, that works but only with chlorine. Chlorine forms a gas and it dissipates out of water. Letting water sit for 24 hours will cause most if not all of the chlorine to dissipate out. Vigorously agitating the water will dissipate the chlorine quickly, which is why some water boards have stations along long water routes to add more chlorine--the chlorine added at source dissipates out as the water travels through the pipes.
But nowadays there is usually more in tap water than just chlorine. Many places use chloramine, which will not dissipate out like chlorine, but instead is connected with ammonia. I am not a chemist and can't fully explain the scientific connection, but the only way to remove chloramine is with chemicals such as in a good water conditioner made for the aquarium.
Tap water may also contain ammonia, in some areas it is very high. Nitrite may be present, and/or nitrate. It is good to test the source (tap) water for all these before first using the water, just so you know the water properties and what has to be done (if anything) to handle it. I would never consider using tap water without a conditioner; even if it is OK today, there is nothing to stop the water supplier from adding something to combat bacteria or whatever, and you may not always know in advance. I speak from experience.
Ammonia at 4 is going to have a negative effect on the fish. They may live through it, but internal damage can occur which may manifest itself weeks or months later. This goes back to what I said at the start; the fish is trapped in the aquarium and anything in or added to the water is going to have some impact. If you had a molly and a platy in an aquarium and the ammonia rises, the molly will almost certainly die within days, but the platy may survive. Mollies are highly sensitive to ammonia, which is why those who add them to new tanks frequently have them die. We tend to label fish like the platy in this example "hardier" but that only goes so far.
Corydoras: I have maintained dozens of species over 20+ years, and I can assure you that they are very sensitive to water parameters and water conditions. They do not tolerate medications or salt. In my view salt should absolutely never be used with corys in the tank, and medications must be very carefully selected. I can honestly say that the fish with which I have had the most problems with respect to disease and health issues have been the corys. They are usually the first to get something, the first to die from it, and the first to show stress and even die from medications. Most of the catfish are like this too.
You asked about conditioners: Most handle chlorine, and now most handle chloramine. So this is basic. Depending upon what you have in your source water, one of these may be sufficient. However, many (but not all) will also detoxify heavy metals, and this includes iron, copper, zinc, manganese, nickel...and any or all of these may be present. Usually in "trace amounts" because these minerals are toxic to humans too, so water boards will monitor and treat their water accordingly. But the level of copper for instance that is accepted for humans--1.3 ppm is considered "OK" for humans--is quite a bit higher than what a fish can tolerate--copper above .002 ppm can kill fish. So metals in the tap water may be higher than fish will tolerate. A conditioner that handles heavy metals is advisable, and most will.
Beyond that, I don't worry because I have heavily-planted aquaria and thus ammonia, nitrite and nitrate is not going to be a problem and I know none of these are in my tap water. But without plants, any of these can be problematical. A conditioner that detoxifies ammonia is a good idea, and several include this property. One that also detoxifies nitrite and nitrate is probably good in new aquaria without plants, and to my knowledge the only one that does all this is Prime made by Seachem. Some aquarists use it regularly; I wouldn't because I don't want all that "detoxifying" in my planted tanks. Given that you are starting a new tank and thus will have ammonia and nitrite issues, Prime is a good choice at least until the tank is cycled. Or if these forms of nitrogen occur in your tap water.
Nitrate is less harmful and most advise keeping it below 20 ppm, and a regular weekly water change of 40-50% will normally achieve this--unless nitrate is in the tap water.
This is getting lengthy, so I'll stop. IF you have further questions, don't hesitate.