No, we don't have a concept of articles, so for us it's not that кіт means both "the cat" and "a cat", we don't even differentiate between "the cat" and "a cat", so кіт doesn't really mean either, it just means "cat" and that's all... So hard to explain :) But if you want to translate to English then yes, кіт can be translated to both "the cat" and "a cat" depending on the context.
When we want to achieve a specification obtained by using "the" in English we just literally say "this cat" (цей кіт) or "that cat" (той кіт), while "a cat" will definitely just be кіт
It can be either a hard "d" sound or a soft "d" sound depending on what comes after.
To make a soft "d" sound, you have to pronounce it with the middle of your tongue raised and conversely to make it hard you need to have the middle of your tongue lowered.
Д will make a soft "d" sound when it’s followed by one of the six letters ь, є, ї, і, ю, я unless there is an apostrophe in between.
I want to caution you against thinking the Ukrainian alphabet is just a slightly different version of the Russian one. Although it might be helpful to keep things straight in your mind, the Russian sound represented by ы does not correspond to the Ukrainian и. They are different sounds (even though neither exists in English as a separate phoneme).
It’s a bug due to the fact that there seems to be some romanization going on behind the scenes. It shouldn’t be accepted. Note also that typing "i" on the English keyboard shouldn’t be accepted either, the "i" from the Latin alphabet is not the same letter as the "і" from the Ukrainian alphabet even though they look very similar.
The difference is that one is a letter in the Latin alphabet and the other is a letter in the Ukrainian alphabet. With the font used by Duolingo there doesn’t seem to be much difference visually, but there might be differences in other fonts.
And the difference in the ASCII (or rather Unicode) code matters. For instance the Wiktionary page about "кіт" in Ukrainian is https://uk.wiktionary.org/wiki/кіт but the following page returns an error: https://uk.wiktionary.org/wiki/кiт
Slight error: You put the same link twice. https://uk.wiktionary.org/wiki/kit (The Latin alphabet letters gave an article on the International Phonetic Alphabet and an article on their best translation into Cyrillic as кит which means whale.) https://uk.wiktionary.org/wiki/%D0%BA%D1%96%D1%82 (The Ukrainian word meaning cat in Cyrillic letters: кіт)
«Кішка» is a perfectly normal Ukrainian word. «Кітка», «киця» and «кіт» should be accepted too, but there's nothing wrong with «кішка».
The Russian word for 'female cat' is «кошка».