1) Cayde wasn't known to "go on missions". Yeah, he went out, but that was for gambling, and whatever shit got in the way. And if he did go on strikes, he usually went alone, textbook hunter. And, if he was partnered with anyone, Eris would have been a dime-a-dozen.
2) Through out the TTK, Cayde pissed off Eris... a lot. Not teasing or flirting, just their characters. Cayde is carefree and sarcastic, Eris is stiff minded and only focuses on the serious threat that is the Hive. Cayde putting on her face was more to continue to piss her off... Still funny.
3) Once again, Eris would be a dime-a-dozen. He literally be talking about any girl from Venus to the Reef; from many centuries past to the present.
English
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I believe number three is talking about the precious hunter vanguard, who was in a relationship with Cayde, She had dies and Cayde took over the hunter vanguard in memory of her.
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The previous hunter Vanguard was Andal Brask, a man.