Why do you think that immediately after praying for a child, Chana said that if her prayer was granted, she would dedicate his life to Hashem (as a Nazirite)?
If Eli HaKohen thought that Chana was doing something wrong, why didn't he interrupt her?
How could Eli have mistaken Chana's heartfelt prayer for drunken intoxication?
Rosh Hashana is the day when we crown Hashem the King. Likewise during the daily prayer, one is to view themselves as standing before the King.
How is it appropriate to ask for one's own needs at such a time?
Why did Hashem create us with desires for food and drink? The secret is that everything has a spark of holiness within, and these sparks yearn to be elevated from within their husk and cleave to holiness. The desire for everything that one eats and drinks stems from the spiritual connection to the food and drink that he needs to rectify. That is what the verse in Tehillim means, "hungry and thirsty," namely, that when a person is hungry or thirsty, this is because [as the verse continues], "their souls within them weakened" [that is, a person's hunger & desire to eat is really motivated by his soul's desire to utilize the food in the service of Hashem and thereby uplift it].