Why is "t" sometimes pronounced like "d" in American English?

First two questions: The pronunciation of some American English consonants can be quite different from British English, in particular for R and T. A t in the middle of a word can be pronunced as a soft d in American English (think of bottle, cattle, etc.). See here, for example, for examples of this.

Third question: Why it does happen for Italy and not for Italian is clearly a matter of stress. If the stress is on the t, it usually keeps its pronunciation and is not changed into a soft d. Thus /ˈɪdəli/ but /əˈtæljən/. Another example is (taken from the New Oxford American Dictionary, in US English pronunciation): tautology (/tɔˈtɑlədʒi/) vs tautological (/ˈˌtɔdlˈɑdʒəkəl/), which clearly demonstrate that.

Not all Americans do, and not consistently.

Flap-t (/d/ instead of /t/) often happens between vowel sounds or after a vowel and before a liquid.

The t in "-teen" is always pronounced as t. As Henry mentions the reason is that flap rarely happens in stressed positions. As it doesn't happen in Italian.

Here's good explanation of T pronounced D.

There's no /d/. It's /ɪtəli/ pronounced [ɪɾəli], with alveolar flap in the pronunciation corresponding to phonemic t. It really is helpful, folks, to distinguish between phonemes and phones for a question like this. If the schwa is lost in casual pronunciation, since you can't say a flap right next to an [l] (the tongue tip has to come back down for a flap), the flap becomes [d]: [ɪdli] (which does not happen in my pronunciation, but I've heard it).