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The и in both of these sentences appears extra. What is it doing?
— Зачем мне насос? — ответил Гена. — Я и ртом надую.
And from another children’s book:
Всех друзей я киселем вкусным угостила, не беда, что мне самой так и не хватило.
Feel free to answer in Russian or English, I can read it, I just can’t write it.
In the first case the closest english word I can think of is “even” like in:
I can inflate it even with my mouth.
The second one is slightly more difficult. «Так и не» is often used as “never”.
Её даже построили, но «добро» на запуск так и не было дано. — It had actually been built, but the fire order had never come.
This meaning actually doesn't seem appropriate in this case, but I (being a native speaker) interpreted it exactly this way. While writing my answer I realised that this is nonsense and the author probably meant something else.
So, here is my guess. «Так» stands for “thus.” «И» here doesn't have any special meaning. You can often find it in sentences explaining something.
Он молодой, потому и быстрый.
Compare to:
Он молодой, потому быстрый.
These statements are essentially the same, but the latter one is more like a logical proposition, saying “He is young. As a consequence he is quick.” The former is more like an explanation: “He is young and that's the reason why he is quick.”
I'm sure it's difficult to grasp. Maybe others, who know theory, can explain it better.
The first one means "just as well" which is also, in a sense, redundant, so your intuition is correct. I can/will inflate it with my mouth (just as well). Often one can also translate this usage of и as "too" or "also", too ;)
The second one is not a standalone word but a part of the compound particle так и which means "in the end" here (there are other meanings too). It can be used without an explicit negative: *всех покормила, а сама так и осталась голодной" I have fed them all but stayed hungry myself in the end.
To me they seem to be two different usages. In the former example:
— Зачем мне насос? — ответил Гена. — Я и ртом надую.
the particle и seems to imply alternatives: "I don't need a pump, because I can also inflate [whatever] with my mouth[, and possibly with my nose...]"
In the latter example the и seems to hint at an omitted clause:
Всех друзей я киселем вкусным угостила, не беда, что мне самой так и не хватило.
"I brewed some kissel for my friends [but they ate all of it] and therefore there wasn't anything left for me".
Её даже построили, но «добро» на запуск так и не было дано.
"It had even been built, but [the management had changed their minds] and didn't give the go-ahead to launch it.
I’m native Russian, but I cannot say exactly what does this и means. As I see, it strengthens the expression. @kirelagin gave good translations and examples. In general, и used like this doesn’t need an exact translation. It will be OK if you first translate sentence without it, and then just strengthen it.
In those sentences "и" means "even", it's used for making the sentences sound stronger or convey the ideas stronger, deeper. It can be omitted but the original meaning will be lost as well.
