Pronouncing Ukrainian vowels: yikes!
When I hear the audio pronunciations of Ukrainian words, the vowel sounds often don't sound like the vowel descriptions written for each lesson. For example, the Ukrainian "a" is said to sound like the vowel in "bus." But in the audios it often sounds like the "a" in "father."
Similarly, when there are two vowels at the end of a word, like "иі" for example, to this American ear it sounds just like the single Ukrainian vowel 'і."
In any case, the various e, a, and i sounds are quite subtle to an English speaker. In our everyday speech American speakers tend to smoosh their vowels into just a couple of sounds for each vowel. Do Ukrainians smoosh their vowels as well?
Note: "To smoosh." American colloquial. To squash, to mash or flatten.
I think you're better off ignoring those comparisons completely. English has a very weird collection of vowel sounds, compared to other languages, and they differ considerably from dialect to dialect. You're best off learning the vowel sounds of Ukrainian, or any other language, by listening.
^^^ this. The comparisons with English words are always going to be dodgy considering how different various English accents can be. Don't worry too much about the "a as in father" type comparisons, listen and repeat, go on Forvo, etc.
About the only time I've ever found it useful to attempt to figure out how to directly map one alphabet to another is in order to practise the alphabet by writing a language I do know in an alphabet I don't, which as you progress in the language becomes fairly obsolete anyway (though it can still be kind of fun!). When you have audio right there, use it, it's going to be vastly more helpful to you in the long run.
Regarding "а", in my dialect of English (West Coast US) I pronounce the "a" in father very similar to the "u" in bus. The difference between them is very small, and maybe where the makers of the course are thinking from.
For the other two points you made, they're called Diphthongs, and Ukrainian does have them. Diphthongs come about to make language be able to be spoken faster. It's probably best if you don't use the diphthongs from the beginning so you can remember vocally how the word is spelled. Just be prepared to hear vowels "smooshed" together.