I have a doubt. Can I use this contraction?
Karen and Tony've got a computer
Instead of the full form of have got :
Karen and Tony have got a computer
Which one should I use? Or, are both correct?
I've never seen the 've (for have) contraction on anything but a pronoun. However the 's contraction for is is common with just about any singular noun or pronoun. In informal speech a little slurring will occur with are as well so that in effect you'll hear Sam and Jane're coming to town: but this is informal speech, and it never gets written out that way.
I don't know about correctness, but I try to write as I speak. So, for me, the issue is whether this is an acceptable English pronunciation: *[kɛɹənəntownivɡɑɾəkəmpjuwɾɹ̩]. And it's not. I say [əv] here, not [v]. And [əv] is a regular reduced form of [hæv] when it is not stressed -- it doesn't depend on a special contraction. So I'd write "have".
A Google Books search yields almost 300 matches for the phrase "Tony's got," where the meaning is "Tony has got." A typical instance comes from Liberty magazine, volume 11, issue 3 (1934):
Tony's got a temper. Also he's got an itching for money, and if he got his fifty for that match he'd have fifty dollars and forty cents—enough, maybe he thought, to take him and Milly back to Yonkers.
So (to answer the question in the header on this page) it should be abundantly clear that you can use a contraction with a proper name as well as with a subject pronoun.
As for contractions of the type "Karen and Tony've got," they are apt to be less common if only because situations involving a compound subject with Tony as the second element are much rarer than situations involving Tony as a singular subject. Nevertheless, the contracted form "Tony've" does occur in some Google Books search results. For example, from Gill Hornby, The Hive (2013):
“Bea and Tony've dumped her in there, sold her house, say she's gaga. But she's always pretty lucid on the subject of Beatrice, I can tell you.”
In short, the contracted form "Tony've" can be used, has been used, and will continue to be used in spoken English and in writing that tries to mimic spoken forms of English. For even slightly formal written English, however, I see no reason to prefer "Karen and Tony've" to "Karen and Tony have."