I am so confused.

[SeaAngel]

Tip: Using a plastic gift card or old credit card will scrape the algae off the glass better than anything. Gkoid luck!

 

[Ash Paws]

I am confused by some of these numbers. Can you test your source water which I assume is tap water, on its own, for nitrate, nitrite, GH (these three especially)?

Here is the test results for the tap water

Nitrate 0
Nitrite 0
pH 7.0
kH 80
gH 120

 

[Ash Paws]

Tip: Using a plastic gift card or old credit card will scrape the algae off the glass better than anything. Gkoid luck!

That sounds like a good trick, I will have to try it.

 

[Byron]

Here is the test results for the tap water

Nitrate 0
Nitrite 0
pH 7.0
kH 80
gH 120

The above is the tap water. For reference, here are the test results posted for the tanks:

Nitrate: 20 ppm (5 gallon), 0 ppm (new 10 gallon)
GH: 120 ppm (5 g), 60 ppm (10 g)
pH: 7.5 (5 g), 7.0 (10 g)

Nitrite and KH were the same for both tanks.

The GH of the new 10g is half the GH of the tap water and existing tank, so this result may be inaccurate for some reason, but we can ignore that.

Nitrate is more of a concern, since the 20 ppm level in the existing 5 gallon is occurring solely due to the fish load. More substantial water changes should reduce this, along with not overfeeding, and separating the neons out as you intend. So that is on the right pathway going forward.

 

[Ash Paws]

Pence is in the ten gallon tank with the tetras. I didn't want him to go in with them, but the water quality in his other tank was very poor. This may sound very concerning, but I put beneficial bacteria in the 10 gallon and today I will get an ammonia test kit.
I did a huge water change of the 5.5 gallon, cleaned out the gravel, and wiped off the algae. I'm letting the water settle for a bit before putting Pence back in. I also put some BettaFix in the water, which helps bacterial infections and kind of relieves stress too. I suspect Pence may have dropsy so I am definitely moving him to the 5.5 gallon.

Also yes, It would have been better to wait a while for the tank to cycle, but Pence's tank and condition was so bad I just had to get the tetras out.
If you look at my other thread there will be more info.