Hi sianeds,
The jelly like egg masses is most likely snail eggs. If you don't take them out, they will be hundreds of snails you will have.
The breeding Neon can be done at home but not in the community tank. You need to set the breeding tank/container(doesn't have to be big. I done it with 1 1/2G ~2 1/2G. Probably could even do with 1G). They are so call egg scatter.
They are considered one of the challenging fish to breed. It is because, the aquarist have to do to hatch the eggs and raise the frys. And they are both small and need certain water and tiny food for them to raise the frys. And it require frequent cares.
If the fish is healthy and tank water is acceptable, they would lay(scatter eggs) even in the community tank. However, they will not survive since other fish or even the fish layer would eat those eggs. Most of them, don't even make it to the bottom. They would be catch and eat by the fish in the mid water.
Not to mentin the eggs and small frys are supposed to be light sensitive.
It is not financially rewarding since the Neons are mass produced in S.Asia and they are like $1 a piece in U.S now(it is hard to believe they were $500~1,000 a piece when they were introduced first time in hobby and that is old money so it worth more in current money). And I tied like once a month, just pull a pair(fat female and best colored male) from community tank and place them in the little container with breeding net or some marbles on the bottom so parents won't eat the eggs. I have a dozen or so free-swimming frys right now. Right after they hatch, they hatch in about day and half after they lay eggs. However, they need really soft water to hatch. Almost 0 hardness so either you need some collected rain water, RO water or distilled water. And the wigglers are so small and almost colorless and almost tough to see with your naked eyes. And another 4~5 days, they just use up their eggsack and when they become free-swimming then you need to start feeding them. But food have to be so small almost you can't see.like infusioa, green water or some micro organism. Around that time, you see the small black pair of eyes swimming in the container. But it is a great experience and sense of accomplishment if you breed Neons. Especially when you see them transform from brownish dull fish gain little red coloring then finally the blue neon line appear after a month from hatching. That is when you see the tiniest Neon Tetra.
And if you can breed Neons, you can almost breed any other tetras maybe exception of Cardinal Tetras(I hear their requirement is more strict. I never done that since I never owned them. Not to mention, some reason I like Neons much better)
Anyway, you can always web search, just type "Breeding Neon Tetra" and you can find some good info.
If you have any question, you can catch me. I have done more done few time. But raising babies is raising babies, it require some commitment from you and time and effort and some care.
But I think it is great experience.