Audio sounds as if the "yr" took the swa-sound from ysgol 's "y" as if it would be a single word. Is it the way it is used or it is just a misheard voice? (Sorry, I don't know IPA and as I am not native English speaker, my attempts for present the sounds could be misleading...)
I can't vouch for whether the TTS is accurate, but I suspect you are correct that the sounds do/should run into each other. I've heard it in other words/phrases as well. I'm assuming it is correct and what is usually referred to as 'liaison'. It happens a lot in some other languages, epecially French.
In fact we use it a bit in English, informally when speaking quickly, but don't really realise/notice.
I'm sure a native Welsh speaker will be able to confirm this at some point.
I would say that there is only one "the" but it takes 3 different forms depending on its surroundings. (A bit like the indefinite article in English has the two forms "a" before consonants and "an" before vowels). There is "y" before consonants, "yr" before vowels, and "'r" after a vowel whatever the next letter is. So "y coleg", "yr ysgol" but "i'r coleg", "i'r ysgol". As for where you use "the" in Welsh sentences, it is similar to English but there are differences in certain circumstances where one language uses the definite article and the other does not. I'm afraid I cannot adequately and concisely explain this - you're better off checking in a Welsh Grammar!