Is चाय pronounced the exact same as if it were spelled चाई (or चाइ)?
No. चाय is pronounced with the 'y' sound (as in 'yam') at the end.
Hindi does not have diphthong vowels. So, in pronouncing something like चाइ and चाई, there is a tiny pause between the two vowels (which is why you won't get the English pronunciation of 'chai' in Hindi). In fact, many native speakers would rather pronounce these as चायि and चायी.
You can see this in words like गई (the feminine past tense form of जाना-'to go') where a lot of people pronounce it as गयी.
I am confused by the lack of dipthongs. So many things are transliterated as two vowels in a row. How are these different to Diphthongs? Some are just one long vowel, like आ as aa, but i've seen ai too? You say you pause in the middle of a and i for chai, but the thing "ai" see it transliterated as in the duolingo course looks like one letter?
As you say, आ is a long vowel rather than a diphthong.
ऐ and औ are transliterated as 'ai' and 'au' respectively but they are not diphthongs in standard Hindi. ऐ is pronounced like 'a' in 'bank' and औ like the 'o' in 'off'.
The transliteration is because these characters are pronounced as diphthongs in Sanskrit, which uses the same script as Hindi.
The Hindi word चाय would not be transliterated (in the ISO scheme that this course uses) as 'chai'. It would be cāy.
'Chai' on the other hand, is an English word which has been borrowed from the Hindi word चाय.
Of course, in casual transliterations that are used in day-to-day life, you'll see all kinds of different transliterations but these assume that you know the correct pronunciation already.
Yes. In Hindi written in the Devanagari script, all vowel and consonant characters have a fixed pronunciation that should be used in all words. A single Devanagari character cannot represent two different sounds (not taking into account regional accent differences).
So, य् is always the sound of y in yam.