How do Wikipedians label articles by their quality level?
Most articles are not labeled at all and correspond to English Start and C. Above they are labeled with a template for that level.
What levels are there and what processes do they follow when labeling articles for quality?
Utmärkt artikel (like FA) with template https://sv.wikipedia.org/wiki/Mall:Utm%C3%A4rkt
Bra artikel (like GA) with template https://sv.wikipedia.org/wiki/Mall:Bra
Rekommenderad artikel (like B) with template https://sv.wikipedia.org/wiki/Mall:Rekommenderad
Stub with a template in category https://sv.wikipedia.org/wiki/Kategori:Alla_stubbmallar (which puts the article in the category https://sv.wikipedia.org/wiki/Kategori:Alla_stubbar )
There is a list of criterias for the three in the top here: https://sv.wikipedia.org/wiki/Wikipedia:Kriterier_f%C3%B6r_utvalda_artiklar
Anyone can label/unlabel articles as stubs or Rekommenderad (except yourself if you are the main editor) following these criterias.
For the top two there is a peer review process.
Worth to note is that most of the stubs are bot created (and of pretty decent quality) and will be placed in sub category to https://sv.wikipedia.org/wiki/Kategori:Robotskapade_artiklar
How do InfoBoxes work? Are they used like on English Wikipedia?
Infoboxes are very similar to English Wikipedia, and we are getting more and more that use Wikidata supported templates with no need for parameters.
Are there "citation needed" templates? How do they work?
Yes, the main one is https://sv.wikipedia.org/wiki/Mall:Kb and is used for each statement.
There are some more templates in this category: https://sv.wikipedia.org/wiki/Kategori:%C3%85tg%C3%A4rdsmallar_r%C3%B6rande_k%C3%A4llor_och_upphovsr%C3%A4tt covering corner cases (like this paragraph needs a source) or more general (like there are to few sources in this article as a whole).
All of these templates put the article in the category: https://sv.wikipedia.org/wiki/Kategori:Alla_artiklar_som_beh%C3%B6ver_k%C3%A4llor