tank maintenance

Keeping the nitrate below 20 is a good rule of thumb. If the nitrate in the 3-gal rises so quickly following a water change, try adding some live plants. They consume nitrate. Although, if it were me, I'd probably still do at least two water changes per week on that tank, because it's so small, even if the plants do help keep the nitrate down.
By the way, what's your stock in the 3 and 12? How many of each fish?

The 3-gallon is overstocked right now while I am struggling with the 12. It has two young mystery snails and a single danio. I think I am going to move the danio to the 6-gallon, which currently has 4 danios and a cory. The mystery snail in the 6 gallon recently died, so I think there is room for one more fish and then he doesn't have to be alone. My original plan was to also move one of the snails to the 12-gallon and then figure out what would be appropriate to put with the remaining snail in the 3-gallon.

The 12-gallon right now has a single huge mystery snail in it. I had a battle trying to cycle it, and then it finally cycled. I started stocking with guppies and they got ick and died. So I am now in the process of cleaning out the tank and I am going to wait until after the holidays to try again. The plan is 4 male guppies and 3 male platys, plus the snail (and possibly a snail from the 3-gallon).
 
The plan is 4 male guppies and 3 male platys, plus the snail (and possibly a snail from the 3-gallon).

I hope the cycles all even out for you soon! Just a suggestion, but think about moving the cory from the 6 to the 12, when the 12 is ready, and then getting a few more cories to live with him. Like danios, cories are social fish that do best when kept in groups. People usually recommend a minimum of at least 6, but your number would depend on the rest of the stock in the 12.
 
I hope the cycles all even out for you soon! Just a suggestion, but think about moving the cory from the 6 to the 12, when the 12 is ready, and then getting a few more cories to live with him. Like danios, cories are social fish that do best when kept in groups. People usually recommend a minimum of at least 6, but your number would depend on the rest of the stock in the 12.

Would I need to put a bottom feeder of some kind in the 6 to replace the cory?
 
Would I need to put a bottom feeder of some kind in the 6 to replace the cory?

Oops! Sorry I missed this from a couple of days ago.
As for your question, I would say, no. There's no reason any tank has to have any bottom-feeders in it.
A lot of people stock their tanks based on this pattern, just for aesthetic reasons: a school of bottom dwellers like cories, a school of mid-water fish like neons or rasboras, and a centerpiece fish like a gourami. This is so that all levels of the tank are occupied and things look active but balanced.
However, you really need at least 20 gallons for something like that, to make sure that you have decent sized groups for your schooling fish (maybe 10 to 15 gallons if your fish are all tiny, like dwarf cories and micro-rasboras). In a 6 gallon tank, there's really not a lot of room for complicated stocking. If you've already got some danios in there, there's no need to add any bottom-dwellers.
 
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