I can only answer parts of your question, but I hope this is of some use before a more elaborate answer appears.

Are they all proper nouns or are there any exceptions with common nouns?

Two words come to mind: fīō and āēr. I cannot think of other examples outside Greek names, but perhaps there are some. These two are notable in that they are common (especially the first one) and perhaps not seen as foreign.

However, these two examples have only two syllables and therefore the length of the vowel before the other one has no effect on the stress pattern. The effect becomes more clear in āëris. With fieri we have prefixed versions and imperfect forms like fīēbat, but length does not change stress unless a prefix can take the stress in a short word.

What is your own experience with such cases? Do you learn (or teach) them on a case-by-case basis or do you deal with them somehow systematically?

The way I see it and would teach it — the closest thing to actually teaching Latin I do is using this site — is that this is not a hard rule but a strong tendency. Exceptions are to be expected in Greek loans and the peculiar conjugation of fieri is the most striking exception. This rule improves intuition greatly but is not infallible.

Why does this exception not apply to Pausanĭas (cf. Παυσανίας)? Why Thalīa (cf. Θάλεια) but Uranĭa (cf. Οὐρανία)?

This is nothing but a guess but something worth considering: metric constraints. Especially names like Urania are likely to appear in poetry, and it simply will not go into hexameter as Ūranīa. Hexameter would explain nicely why Pausanias and Urania have a short I whereas Thalia has a long one. Surely more data points are needed to see if rhythmic considerations like this correlate with the observed phenomenon. (It could also be that some names are slightly modified to fit metre, and our knowledge of quantities comes from that biased sample.)

I can only answer parts of your question, but I hope this is of some use before a more elaborate answer appears.

Are they all proper nouns or are there any exceptions with common nouns?

Two words come to mind: fīō and āēr. I cannot think of other examples outside Greek names, but perhaps there are some. These two are notable in that they are common (especially the first one) and perhaps not seen as foreign.

However, these two examples have only two syllables and therefore the length of the vowel before the other one has no effect on the stress pattern. The effect becomes more clear in āëris. With fieri we have prefixed versions and imperfect forms like fīēbat, but length does not change stress unless a prefix can take the stress in a short word.

What is your own experience with such cases? Do you learn (or teach) them on a case-by-case basis or do you deal with them somehow systematically?

The way I see it and would teach it — the closest thing to actually teaching Latin I do is using this site — is that this is not a hard rule but a strong tendency. Exceptions are to be expected in Greek loans and the peculiar conjugation of fieri is the most striking exception. This rule improves intuition greatly but is not infallible.

Why does this exception not apply to Pausanĭas (cf. Παυσανίας)? Why Thalīa (cf. Θάλεια) but Uranĭa (cf. Οὐρανία)?

This is nothing but a guess but something worth considering: metric constraints. Especially names like Urania are likely to appear in poetry, and it simply will not go into hexameter as Ūranīa. Hexameter would explain nicely why Pausanias and Urania have a short I whereas Thalia has a long one. Surely more data points are needed to see if rhythmic considerations like this correlate with the observed phenomenon. It could also be that some names are slightly modified to fit metre, and our knowledge of quantities comes from that biased sample. Modifications due to metric constraints may have happened in Greek before loaning to Latin.