Yes, we are all trying to give you the benefit of our experiences, losing fish, having sick fish--many of us went through it because when we started (at least when I did back when I was in school and that was decades ago
) there was no internet with forums to go for advice, and we learned by failing and lost many fish along the way.
Anyway, you need to decide what you want first, and then work towards it. If you really want Lemon Tetra, fine; as I said, get 4 more so you have 6, and they will manage in your 10g. When you can afford a 20g, you can move them in there with another species, like neons (6 of them). And perhaps 5 corys for the lower level interest, or something else; corys also need to be in a group, 5 works well and is workable in a 20g tank.
On the other hand, if the 10g is it for the present, then select fish that will be happy (= healthy) in a 10g. Ember Tetra work fine, a group of 7-9; there are other small fish, have a look at the dwarf rasbora, Boraras brigittae, in the profiles; or the Scarlet Badis, Dario dario.
Livebearers were mentioned; the Embers and other fish above are soft water fish, while livebearers are harder water fish. Nothing has been mentioned yet about your tap water parameters (hardness, pH). This has a bearing on which fish are compatible.
A comment on filtration; most of these fish prefer less water movement. In a 10g and 20g tank, an ideal filter is a simple sponge filter connected to a small air pump. Something else to think about.