This isn't normal Scottish English. I stay in Glasgow. No Glaswegian would ever say that. I could just about accept I am staying in Glasgow if it is temporary.
It is notable that stay is the direct translation of fuireach but no one knows which came first. There is a preference on this course for English that corresponds with the Gaelic.
It is a rule that applies generally, to all verbs, so any verb where the simple past and past participle differ in English will do as an example
A've did (I in English) etc. etc.
The only example I can think of where the word is not found in English at all is
They've jamp / they jamp (Scots)
They've jumped / they jumped (English)
The "r" in "fuireach" sounded like a "d" to me. Is this how it is pronounced or did I not hear it correctly? Another observation is that the cluster "sch" in "Glaschu" was pronounced more as "suh-chu", i.e. a schwa was added. Is this how the conjunct consonants are pronounced in Gaelic?
R Yes you heard it correctly. In most dialects, r is not too different from in English (but very different from Scots), and you cannot tell the difference between a broad and a slender one. However, in some dialects, I think in Lewis, the slender r is different. It is somewhere between a very soft d and an English th. It takes getting used to but it does help distinguish the broad and slender r.
Glaschu Pronouncing pairs of consonants is difficult if you are not used to it. The classic example is the Irish pronunciation of film which is pronounced filim to make it easier for them. This is most common in Gaelic after an r, e.g. gorm is pronounced gorom, but it also happens elsewhere such as in Glasachu. It is called an epenthetic vowel, or a svarabhakti. Some people claim that the vowel chosen is always the same as the previous one, but it always sounds quite neutralized to me (that is, more schwa-like, as you observe).
Yes. That is because we are not meant to be translating into Scottish English. It is Duolingo policy that we use American English. My point is that the person who would use this sentence, that is someone from Glasgow, would not use American English. The course creators don't know much American English (by their own admission). I assume they generally come from the Gaelic-speaking areas. In the past they would not have known Scots or Scottish English either as they would have known only the Gaelic they spoke at home and the sanitized English (i.e. with any hint of Scots removed) they they were made to speak at school. Now that attitudes have changed and Scottish English had changed from an embarrassment to part of national identity and is frequently heard in the media, it is gradually spreading. But I do not stay in the Gaeltacht, and I usually speak Gaelic when I go there, so I do not know if they have this usage.