Fish bowl is no place for fish. They are all for marketing, but not for fish.
To keep any kind of tiny fish, a minimal 5 gallon is required.
Consider you have a goldfish, which produce tons of ammonia.
You must get a minimal 20~30 gallon tank immediately. A filter rated much higher than 20~30 gallon is needed. That AquaClear 20 is not all that powerful consider you keep a goldfish. Get a AquaClear 50.
Nope, old tank water holds next to 0 beneficial bacteria required for aquarium nitrogen cycle. It wouldn't do you any good even if you use 100% water from the pet store. In fact, I would avoid it, since many pet store especially chain stores have parasites and diseases in their tanks.
What you need is a piece of old filter media, that is where most nitrogen cycle bacteria live on. They only colonize surface areas in your tank, and the filter media has most of it.
Read details on aquarium nitrogen cycle.
What water conditioner are you using? At this stage, I would recommend you to get Prime. It detoxify ammonia up to 48 hours, turn it into less harmful ammonium.
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