I wish I had that water, my discus would love me!
The neons will be fine in that water, but in my opinion the tank is a bit short for them. You might want to consider rasboras as they are less "greyhound" fish than tetras.
It's be interesting to know what species of barb you have. If it's quiet peaceful barbs like rosy barbs then you could simply up their number to 8 instead of getting rasboras. If it's tiger barbs, I wouldn't keep them in a group smaller than 10, and they get way too big for that tank.
Guppies like hard water, so I'd rehome those two.
Regarding the black moors, they are a morph of goldfish, and as such will quickly outgrow that tank. Also as already mentioned they are temperate fish, not tropical, so they shouldn't be kept with any of the others.
If you prefer to keep the black moors I'd consider 150 liters as the bare minimum, but 250 would be probably better. They really do get big, and live a long time, much longer than cats or dogs.
If it was my tank I'd set it up as an asian swamp: lots of plants, a single betta, a group of small rasboras (like Trigonostigma espei for example). A few snails to keep the place clean of left-over food and debris.
Further down the line, once the tank is well mature and you have more experience add small group of kuhli loaches.