Pretty much – you can use it with direct object in genitive, but it's very rare in modern Polish and feels archaic/poetic/bookish.
In modern Polish czekać is mostly used in this combinations:
- czekać na cud (often nouns, here kogo?co? - dativ) - waiting for a miracle
- czekać aż dam znak! (some phrase with verb) - wait till I give a signal!
- czekać do północy [time or some event] - waiting till midnight
There is also somewhat obsolete (but not that badly):
- czekam cię(kogo?czego? - genitive) - I'm waiting for you
Like in this song Link
If czekać na took dative, wouldn't this exercise be "czekam na nowemu telefonowi"? I would think this is accusative
No, it wouldn't work. Mostly it's used for people, like in 'prezent dla X' = a present for X, 'nagroda dla X' = an award for X, and so on.
Also, maybe it's easier to treat such phrases as a whole and not necessarily wonder about 'why'. You wait 'na' something, that's just how it is :)
Hmmm... I don't know what I would say in such a situation. I guess I would just say 'czekam' and not precise what for and why...
Mam pytanie, dlaczego nie mogę pisać "I wait for MY new phone"? Czy to tylko z osobami w rodzinie?
With family members, it's often clear by context that you're talking about your own family members. (Also in English: if you're talking about 'mom', you often don't mean the mother of the person you're speaking with)
With objects, like phones, this is not the case, just like in English.