To an American ear, it sounds awkward, but in British English, this is not uncommon.
Ironically, a Brit will probably tell you that the correct form is "I have
got a small dog".
english.stackexchange.com/users/21/Peter Eisentraut
This is definitely an American English/British English thing, as you can't do it in American English but you can in British English.
In American English, you can't contract "have" if you are using it as a plain (not a "helping" or "auxiliary") verb.
"I've a dog" and "They've a great time" are not grammatical in American English.
There are a number of other restrictions on contractions of "have" besides the one you cite.
For example, you can't use contracted "have" followed by "not": "I've not been there" is not grammatical in American English even though "I've been there" is—if you want to contract, you have to say "I haven't been there".
I discussed this in a question about I’ven’t .
english.stackexchange.com/users/39/nohat
I think what you feel uncomfortable with is contraction of "have" as a main verb.
When it's an auxiliary verb in, say, a perfect, contraction feels fine:
I've had a car before.
But contraction of main verb "have" meaning to own or possess feels weirder. ?
And I've a car right now.
However, I have a feeling that people will contract main verb have in British English, but take that with a grain of salt.
Americans faced with some kind of strange usage are far too willing to blame it on British English.
english.stackexchange.com/users/257/JoFrhwld
I think "I've a car" is fine, but unusual on its own: as part of a longer sentence it's unexceptionable: "I've a car in the garage".
I suspect this is for prosodic reasons: "I've a car
" has no word you can stress, other than "car", so people tend to change it to either "I have a car" or "I've got a car".
The other case is different: I can't think of any examples where I would expect to find "he's a pen", though I would rate it as perfectly grammatical.
english.stackexchange.com/users/547/Colin Fine
"I've" seems fairly normal to me as British (or at least Scottish) English, and is completely unambiguous.
I don't think this
"he's" would ever be understood as "he has" rather than "he is".
english.stackexchange.com/users/4335/neil
They are both strictly correct but both slightly inelegant.
The third especially could be confused with "He is a pen" Some people like to avoid contractions like that in formal writing, but most people probably won't mind (or even notice).
It's probably wise to favour the contraction if you want to emphasise another part of the sentence: "I've never been so insulted!"
english.stackexchange.com/users/117/Mark Pim
It is rarely appropriate to use contractions in a formal writing environment.
I've heard the first construction in speech, mainly British English, but not the second.
english.stackexchange.com/users/7/Michael Scott Shappe
Technically, there's nothing wrong with it, although Uncle Mikey is correct that it's rarely appropriate to use contractions in formal writing.
The only reason the second one seems strange is because most people would say "They're having a great time," I suppose because they're in the middle of having it.
I have said, though, "I'd a great time...
" I just happen to speak too quickly.
:)
I also say "I've got..." rather than "I have.
" When I'm not speaking incorrectly, I generally just say "I've."
So, it's not just a British thing.
english.stackexchange.com/users/23/kitukwfyer
(I've) is a fine contraction, just in American English you can only use it to replace (I have)
when 'have' is used as an auxiliary verb (eg: In conjunction with a past participle).
I've been there.
(Correct)I've a dog.
(Incorrect)
When the verb 'have' isn't auxiliary, It can't be contracted with pronouns.
english.stackexchange.com/users/380027/Syed Wasif Ali
I feel uncomfortable saying sentences like the following:    " I've  a car" instead of " I have  a car"  " They've  a great time" instead of " They have  a great time"  " He's  a pen" instead of " He has  a pen"  etc   I ask this because I read this sort of thing in a book.  Are they correct? And what is the rule? Can you use such forms in a formal setting?
Is it appropriate to use short form of "have" ('ve) when it means possession?
english.stackexchange.com
<verbs><contractions><auxiliary-verbs>
english.stackexchange.com/questions/8
english.stackexchange.com/users/13/serg
How acceptable the short form of have is may possibly be down to region, with this sound being potentially more acceptable in British English than American English. However, regardless of this possibility, it is rarely acceptable to use contractions in formal writing. Views vary but is appears that the contraction indicating possession is used only in UK English and only in informal contexts.
How acceptable the short form of have is may possibly be down to region, with this sound being potentially more acceptable in British English than American English. However, regardless of this possibility, it is rarely acceptable to use contractions in formal writing. To an American ear, it sounds awkward, but in British English, this is not uncommon.  It is not technically wrong to use contractions, but is rarely appropriate to use contractions in a formal writing environment.  
 Although not strictly ungrammatical, this can sound awkward and inelegant. However, this may be heard at times. In addition, it is rarely appropriate to use contractions in a formal writing environment. It appears that this use of the contraction is limited to British English but in any case is still slightly inelegant. The example of "he's" for "he has" is incorrect however, and in any case, contractions are never appropriate for formal writing. You may want to use a contraction in informal speech to emphasise a different word in the sentence.
Not in US English but's it's ok in UK English. Both forms are fine. "I've" is a normal form in British English. Both are correct, but inelegant. They are both current forms but avoid using either in formal writing.
To an American ear, it sounds awkward, but in British English, this is not uncommon.   I think "I've a car" is fine, but unusual on its own: as part of a longer sentence it's unexceptionable: "I've a car in the garage".    It is not technically wrong to use contractions, but is rarely appropriate to use contractions in a formal writing environment.  
It seems to be the case that this usage of the contracted 've is acceptable in British English but not in American English. Contractions are rarely appropriate in formal writing. You may want to use a contraction to emphasise another word in the sentence. The example contraction of "he's" instead of "he has" is not considered correct. This use of the contraction is considered correct but not particularly elegant.