is there a difference between дать и пусть meaning ¨Let¨? for example, can I say: Пусть она сядешь
(note their usage, by the way: «дать» + infinitive / «пусть» + 3rd person non-past form)
"Дать" explicitly means to "allow" or to not prevent, to not keep something from happening. When you make it imperative, you "command" the listener to let something happen.
"Пусть" is a particle used at the start of a sentence and making a 3rd person imperative (which is not directed at your listener):
- Пусть она сядет (or "сидит") = Let her sit (or "keep sitting").
- Пусть он больше никогда не придёт. = Let him never ever come again.
- Пусть будет так. = So be it.
- Пусть будет свет. = Let there be light (also, more authentic "Да будет свет!")
Дай is more of an explicit command.
Пусть means that you want something to be in some way without actually telling the listener to make it so (in this meaning, the particle is only used in the sentences about actions performed by 3rd person subject).
"Slightly" is an understatement. I suspect even non-native speakers can hear съесть / сесть do not sound the same even if they cannot repeat it exactly. Try it: Дай ей сесть vs. Дай ей съесть что-нибудь
In principle, Дай ей съесть can exist but only as an odd sentence with an omitted object (e.g. "I made", "Give me"). Съесть is a verb specifically used to express consuming something completely, so it usually has an object around. If a girl says Я съела банан it means there banana is not there anymore.