How would I ask someone a desire in Japanese?

Question:

I know how to express desires I want. There is either hoshii, base 2
+ -tai, or if you want someone to do something for you, base te + hoshii.

What really confuses me though, is asking someone what they want to do.
Japanese uses politness and discipline, so to ask someone what they want to do, you put the question in negative form. The problem with that is, I'm not sure how to put it in negative form.

Example: Tabemono wo tabemasen ka?
Do you want food?
Why wouldn't it be: Tabemono wo tabetakunai ka? Since adding tai makes it an i adjective, and kunai is the negative form?
Also, let's say I want to take a walk, or play a game.
Is it: sanpo ni ikimasen ka? Or sanpo ni iktakunai ka, for taking a walk?
For playing a game, is it: Gemu wo yarimasen ka? Or Gemu wo yaritakunai ka?

All advice and comments are appreciated, thank you for taking your time to read. :)

Answer:

- Do you want food? (Do you want to eat something?)
I'd say "nanika taberu?" (nanika tabetaku-nai?)

"nai" here, is not really negative sense. it's more like "what do you think?"
e.g. "kare, sugoi to omowa-nai?" "Isn't he great?" (He's great, right?)

Note: we rarely say "tabemono o taberu" ("..... o taberu" or "mono o taberu")
e.g. you see a girl eating food (you'd say "anoko, nanika tabeteru")

- I want to take a walk, or play a game. "AND you want someone to join you"
"sanpo ni ikimasen ka?" or "sanpo ni ikanai?"
"gemu yaranai?" ("yarimasenka" sounds too polite because you usually play games with your friends)
------- to add:
I got your point. yeah, I'd say "that's Japanese", or let's take it one of "idiomatic expressions"... Otherwise, these things can drive you nuts.
I know you want to say "sanpo ni ......-tai ....", for example, but when you said like
"sampo ni ikitai desuka?"
it's ok, and most people would understand that you want to go for a walk and they're sort of invited. But to native speakers it doesn't sound natural, and to be more precisely, "sanpo ni ikitai desuka?" actually sounds like "Are you sure you want to go for a walk?" or "you seem to want to go for a walk"