Seeing the text "Can I've a dog" Is simply wrong... I don't mean technically... Probably perfectly reasonable sentence. But, In speech there is a definite H in there eluding to "Can I-h-v a dog" Which would be written "Can I have a dog". I'm 45 years old and have never seen in written form "Can I've ...." Again. I'm sure its technically correct. But, I've never seen it, except in sentences like "I've never seen it" Anyone else think it looks odd? Probably just me :)
"Can I've" sounds completely wrong to this native speaker (USA variety). But I can't think of an example where "have", when indicating possession, is ever contracted to 've. Maybe the contraction only occurs when used as a helping verb, such as "I have seen" -> "I've seen".
British English actually does allow the non-auxiliary "have" to contract:
"I've a question."
(though even for the Brits you can't contract it when you have an inverted modal like "can"... I think the reason is that this is not the finite present "have" [which would be "has" if the subject were 3rd person], it's the bare-infinitive "have" because of the modal)
I see the distinction you are making, and I agree that there is a difference, although I'll beg to differ about the verb not being contracted. "can't" is the contraction of "cannot", a negative verb, similar contractions being "don't" (= "do not") and "won't" (= "will not"), etc. Clearly in the latter case there's no question of only the "not" being contracted, and in fact "not" can't be contracted by itself, outside of this sort of negative verb construction. But you are right that it is a different situation from the contraction of "have" to "'ve". You are citing a general rule that contracted verb forms aren't used in questions, but I wonder what other examples of such a rule you are thinking of other than 've? Hmm, perhaps: "I will" => "I'll". The contracted form can appear in a statement, but not in a question. Ok, we've got 2 data points, maybe we've got a theory! (Science joke.) Wait, try this: The standard way to convert a statement into a question in English is to reverse the Subject-Verb order, and this applies to the first verb, in the case of a multi-verb construction. E.g., "I can go." => "Can I go?, "You do think that ..." => "Do you think that...?", "We have seen...." => "Have we seen...?", "You will run." => "Will you run?" In the last two examples, the contraction could be made in the statement, but not in the VS-ordered question: "we've seen", "you'll run". So it may be correct to say that helping verb contractions can't appear in questions, because of the SV => VS inversion. What do you think? Of course, helping verbs are the most common contractions, so this version is almost as strong as your original statement. But it also ties in my previous point about the contraction for "have" only happening when it's used as a helping verb.
The linguists I trust most have argued pretty convincingly that the negative forms aren't really contractions at all: they're just a short list of (like 15 or so) negative lexemes. They don't follow the same rules as the verb contractions, several have irregular forms, and you can't really stick -n't onto a new word willy-nilly (unlike -'s, -'d, -'ll etc).
By placing собаку in accusative case, it is made the object of an unspoken verb - which is (according to other comments elsewhere) usually understood to be "to give" or "to have" -> "May/Can I have a dog?"
If it were собака, then your construction would be a logical inference.
One infers the verb from the presence of the direct object. The presence of an indirect object in Dative case doesn't detract from this inference.