I've had some Slinkys for about a week. There were nice and shiny and silver and now they are already oxidized and gray. Is this normal?
Pretty much. I just broke a string today, had the strings on for about a month.
Yeah, I find earnie balls dont last that long, Rotosound are quite long lasting , Ellixer too from what i know. um, Only advice i can give is wipe them after playing, and keep out of sunlight. mine usally last 2-3-4 months.
So I guess the Elixirs are worth it - unless one breaks! (which happened on my 12-string)

I can't remember who makes them now, but someone makes coloured, coated strings. I imagine they'd last a while.
Jackson Kelly KE3 w/P-Rails; Aria Pro II TSB-350
• ZVex Fuzz Factory; DigiTech Whammy; Danelectro Fish & Chips; Marshall ED1; Fulltone OCD; DigiTech Whammy; Marshall JMP1; Boss CE3; Vox AC4TV

EMU 0404; Reason/Record; M-Audio AV40
Maybe I just got a bad batch, but the first and only time I tried Elixer strings, they tarnished. Badly. Quickly. And then the coating started to peel. It can't have been that I just didn't take care of them, because I treated them the same way I do every time, and I've never had this problem with any other brand of strings.
Earnie Ball strings are good but dont last that long, the only ones which are worth buying are the beefy ones, they seem to last a bit longer.