Depending on the flavor, Americans might call them _Fudgesicles_ or _popsicles_ or _ice-cream bars._

American children would not use any variation of _I strongly demand a ..._

These would be common, from more polite to less polite:

Polite:

* May I have a popsicle.
* Would you get me a popsicle.

Neutral

* I want a popsicle.
* I _really_ want a popsicle.

Neutral, but so, so annoying:

 * I _neeeeeeed_ a popsicle.   `[Thanks, Chronocidal]`

Impolite

* Get me a popsicle.



....and, of course, adding "please" onto any of these would make them more polite, while adding _now_ would do the opposite (unless there is contextual reason for it: e.g. _I have finished my homework.  May I have a popsicle now?._)