GREAT observations -- careful and detailed. For someone who doesn't know anything about linguistics, you have an excellent ear. In fact, as it happens, Hungarian and Finnish are actually in the same language family, the Finno-Ugric family, and that's why they have the commonalities you noticed. That family also includes Estonian and the Lappish languages. "That funny way of pronouncing the letter T" is that it is not aspirated, as linguists say -- that is, it doesn't have that little puff of air after it. The technical explanation is this. With that type of sound (a "stop"), you start by building up air pressure behind your tongue, which is closing off the airflow in your mouth. To make the sound, you move your tongue tip down and allow the air to flow through. The difference is that with an English T such as the one in "top", you release the airflow a split-second before you start vibrating your vocal folds, and that's what makes the puff of air. To say an English D sound, you start the vibration before you release any air; and to say that "funny" T you start the vibration at the same time.
Also, English and Hindi are indeed distantly related, but perhaps not as distantly as you might think, as they are both members of the Indo-European family of languages (of which Finno-Ugric is a sub-family), and they do in fact have most of their sounds in common, if not the "melody" (technical term: "prosody", pronounced PRAHZodee).
See the Wikipedia article for more information: