Yep, as Shady_arc said, we consider it still to be the same letter, d, but the sound it produces influenced by the "i" after it is palatalized, a little softer. When you say the hard d (as in да до ду де ди) you touch your teeth with the tip of your tongue (you can even make it stick out a bit as in "th" for an exaggerated effect), and for the soft d (as in ді дя дє дю дь) you press your tongue against the roof of your mouth.
Yes, it makes sense in Ukrainian. Ukrainian does not have any type of article, and the verb "to be" is not usually translated in Ukrainian.
Where is the house?
Is - (a conjugated form of the verb "to be") (not translated)
The - definite article (not translated)
House - (translated)