| | Re: "I got it"
I got it (or I've got it) used in this sense really means You don't have to worry about it - I'm dealing with it. - or in other words I've already taken charge of it Hence the form of the verb, which is not actually a past form, but present perfect with the meaning past action with present result, the same as in eg Help! I've cut myself! or The taxi has arrived.
An example :
A : David, the phone's ringing! I'm in the bath!
B : OK, I've got it!
So it's not really used to say that you want to do something yourself, but rather to agree to help another person.
Very often though, the pronunciation of have is so much weakened that it disappears completely - with the result that it sounds exactly the same as a past verb : I got it. But it's not - as you said, a "pure" past wouldn't make logical sense here.
Many native speakers however, would be convinced that it is an "ordinary" past form. It's the type of expression that is learnt and used as an unanalysed "chunk" - a bit like "Never mind" or "Go ahead". When these expressions are used, the speakers aren't thinking of the meaning of each individual words. They're just "ready made" expressions which are used in specific situations. Ask someone what "never mind" means and they won't say "do not get disturbed at any moment in time" but "it doesn't matter".
In the same way "I got it" is likely to be used as a chunk, with no thought to the actual meaning and therefore the grammar that's behind it. It's a tendency that started with US English, but I think you're now just as likely to hear "I got" as "I've got" in British English too (I say it myself), even when the concept expressed quite clearly needs a present perfect because it's an event which combines past and present rather than being pure past. It's happening with other verbs too, and I think there's strong evidence for the argument that the present perfect is slowly dying out.
A pity - I'm rather fond of it ...