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?