"I would want" is "Eu quereria" (futuro do pretérito - "future's past"), that one may say as "Eu iria querer". "I wanted" is be "Eu queria" (pretérito imperfeito - "imperfect past"); "Eu quero um livro." - "I want a book." - "Jag vill ha en bok."
"Jag önskar" is "Eu desejo/Eu gostaria de", and, in english, "I wish/would like"; "Eu desejaria/gostaria de um alce." - "I wish for/would like a moose" - "Jag önskar sig en älg."
I hope that is correct. :)
It does men she wishes it for herself. You can wish "feelings" for other persons, like "jag önskar honom en trevlig resa"-"i wish him a pleasant trip". However, you don't normally say "jag önskar honom en häst", in that case you probably say "jag hoppas han får en häst" - "i hope he gets a horse".
I see you're trying to draw similarities to the use of "till" in "ägg till frukost" (eggs for breakfast) but these are different contexts, and while in English the grammatical constructions are the same (in regards to the use of "for"), the inderlying ideas are what need to be conceptualized differently in order to create sensible Swedish translations. Simply practicing the different uses of prepositions in various contexts should help. Not a native Swedish speaker, but this is my perspective.
Nothing wrong with the above as such, but a more common construction would be, she wants a horse, but if you want to use wish, then, she wishes for a horse, might be more common, or, a perfect construction, she wished she had a horse. I can't help reading, she is wishing for a horse, and, imagining that the poor girl is trying to do the wishing for a some old nag who has forgotten how to wish! Further, as far as I understand Swedish, not far in the slightest, the Swedish is not verb + ~ ing, but just verb, so no wishing but to wish, wishes....etc