Do luck stat have any effect on the rarity of the item in vendor shop and battle area chest?
Item vendor yes. Not sure of battle chest.
Bought the 500k mythsone/crystal and it spit out ultra rare types sometimes. Legend gears sometimes become crystal or eternal maybe because of eternalize/crystalize set.
I suggest after maxing your farm gear buy all legend and crystal/mythstone. It increases your dust farming speed and crystal loot. Saves money and time too.
Yes luck increases item rarity. Better get Nadroji’s to get rarer legends.
I don’t know 
Yeah for vendor. Have treasured and fortunate perks on your main character and hire your higher level non farming character or ascending character to make the purchases
Arena… No
Don’t buy legends from vendor unless you want legends. Lower items can be crystal or eternal. I’ve tested
Legend items from vendor can also be eternals
Lol I stand corrected 
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I bought a gauntlet one time, and it was a level 1 Eternal Golex’s Gauntlet. I loved the 5 blasts coming out. 
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Ikr. I remember getting level 1 Eternal Fabled Blade. Back when it used to be possible to get high floor items in any floor at any level .
Prob affects, as drop modificators such Eternalized seems to work on shop. About arena chests, you won’t get anything better, since rare chests gives rare items, epic chests epic items, etc.
Was able to get Insolence (defiant) which was suppose to be loot on flr 350 onwards from farming flr200. It save me dust from the temptation of buying its recipe in codex.