To answer the original question, it's Present tense, and the verb construction is Perfect. Together they're often called "Present Perfect", which is a tense in Latin. but only a construction in English.
Yes, have got is an idiom; but that explains nothing except its irregularity, of course. And it's a different idiom in UK English than it is in American English, where it contrasts with have gotten. Like most idioms, where it came from is a long tortuous story.
Get means come to be or come to have, as in
- He got tired. ~ He became tired.
- He got his orders. ~ He received his orders.
(I've always been amused that in German
the verb bekommen means receive but not become.)
In the case of have, especially, if one comments on the acquisition
of something, the implicature is that one still has it -- otherwise,
one would say something different. So the present perfect of get
naturally implicates the present of "have", leading to the equivalence
of have got and have.
The Present Perfect construction uses the auxiliary verb have/has,
plus the past participle of the matrix verb:
The past participle of get is got or gotten in the US; UK
mileage may vary. There is a principled distinction between the two,
since get -- as the inchoative form of both be and have -- is itself an auxiliary, and got has come to have its own
usages in American English, leaving the simple Past Participle slot
to be filled by gotten.
As McCawley points out, one of the functions of the Perfect
is to report past actions still relevant in the present; thus,
reports a past event (catching the cold) which is still relevant
(having the cold), and, since pragmatically what we're interested in is
the present state, I've got a cold is used more often to warn people to
duck when I sneeze than to comment on the events of the past week.
But there's more. Both be and have are already auxiliary verbs, and
are used in many constructions, like Passive or Perfect. Since get
can implicate be and have in some cases, it's been generalized to
substitute in others, where their use is grammatical instead of
meaningful, like the so-called Get-Passive
- He got arrested. ~ He was arrested. = s.b. arrested him.
or in the periphrastic modal have to meaning must
- He's got to go. ~ He has to go. = He must go.
(frequently spelled gotta, because the /v/ or /z/ in /ðevgaɾə/ /hizgaɾə/ is usually inaudible)
or simply, wherever one might use have
- I got a new DVD. ~ I have a new DVD.
Quite frequently children generalize this equivalence to produce
in effect, inventing a new verb because the old one has worn out.