I am not sure that the comment about Gothic churches is correct. The ceilings of Gothic churches were usually ribbed vaults, above which there may have been trusses to create planar surfaces to support roof tiles. Ceilings did not usually incorporate windows or skylights, unless they were domes or squinches. Windows were formed in vertical elements such as walls and drums.
Whereas walls in the earlier Romanesque architecture were massive in order to confer adequate structural stability by counteracting the lateral thrust of the rounded arch roof vault by gravity alone, and therefore could not accommodate large window openings which would entail a significant loss in weight, the invention of the column, buttress and infill wall construction in Gothic architecture meant that the weight of the wall was no longer a constraint, and a window could be made as large as could be accommodated between primary structural members
In some countries windows this large would have let too much light and heat into the interior, where a dark ambience was considered more holy and conducive to prayer. Therefore stained glass windows would have a been a good technical solution to the problem of producing large windows that didn't let in too much light. Given the availability of the technology, it would have been but a small step for the creativity of man to come up with the idea of using stained glass of different colours to create religious images of such beauty that they could express and reinforce the congregation's love and awe for their faith, while depicting important Bible personages and scenes for their edification.
Stained glass windows have some things in common with mosaics, but are composed of much larger elements and need to satisfy structural requirements like adequate wind resistance. So although prototypes probably go back into the mists of antiquity, the art and technology of stained glass windows reached its apogee in the era of Gothic architecture.
In answer to the specific questions, Gothic architecture was developed in lands where at the time there were no denominations of Christianity apart from Catholicism. The main centres of the development were France, England, and German-speaking lands. Following the Protestant Reformation there may have been greater usage in lands were Gothic architecture was preferred, namely in northern Europe, where Protestantism had to a large extent displaced Catholicism. As far as I know, there is no obligation to incorporate stained glass windows in church architecture. Stained glass windows are a decorative and artistic choice. There may be some Protestant sects that eschew stained glass windows on the basis that they are figurative representations like paintings and sculptures deemed to be contrary to a literal interpretation of the commandment about graven images.