Where does “coffeehouse” come from?

I need to work through this chain carefully, but I notice there are some issues with the provided etymology chain that make it impossible to write an accurate sentence. The chain shows: - English house | Dutch koffie | Italian caffè | Arabic قَهْوَة | Proto-Indo-European ne- | Proto-Indo-European éǵh₂ | Proto-Indo-European éǵh₂ | Proto-Indo-European en Problems: 1. This chain appears corrupted or nonsensical - it jumps from Arabic قَهْوَة (qahwah, meaning coffee) directly to Proto-Indo-European roots (ne-, éǵh₂, en), which are unrelated language families that wouldn't connect to Arabic etymologically. 2. The PIE roots listed (ne-, éǵh₂, en) don't appear to connect logically to "

coffeehouse (English): An establishment where coffee is served to...

Definitions

  1. An establishment where coffee is served to...

Ancestry of “coffeehouse”, step by step

coffeehouse traces back through two separate lines of ancestry.

via English coffee

StepLanguageWordMeaning
1EnglishcoffeeA beverage made by infusing the beans of the...
2Dutchkoffiecoffee
3Italiancaffècoffee; a café
4Turkishkahvecoffee
5Ottoman Turkishقهوهcoffee
6Persianقهوهcoffee
7Arabicقهوةwine; coffee; coffee shop, café

via English House

StepLanguageWordMeaning
1EnglishHouseA particular chamber of political representation
2Middle Englishhoushouse, residence; house of worship, temple;...
3Old Englishhūsdwelling, shelter, house; house
4Proto-West Germanichūshouse
5Proto-Germanichūsąhouse
6Proto-Indo-European(s)kewH-to cover, bedeck; cover
Every word from Arabic قهوةEvery word from Persian قهوهEvery word from Ottoman Turkish قهوه