What is the status of 什么来的 and 什么来着 in standard and colloquial Mandarin Chinese?
I agree with your friend. I think the correct version is "什么来着". You can find the word"来着",but you can't find the word"来的“ in the dictionary.
I think it's popular in northern area(such as 北京，天津，河北，辽宁). I have heard of "什么来着" on the TV and I can understand it, though I have never used "什么来着", either in mandarin or in my dialect.
(Not applicable)
Of course the dialect will influce the way you speak mandarin Chinese, both in pronunciation and words. Examples from one of my roommates when I was in the university, he came from 浙江 province(in south China), and his mother tongue is a subclass of 吴 dialect.He likes to use words below and confused me at the first time when I met him, though he speaks in mandarin(not very standard).



  纸头 paper  In mandarin, we use 纸 or 纸张
  烧饭 to cook In mandarin, we use 做饭
  A structure "不要太[an adj]". He means "very, quite [adj]", but in mandarin, it means "don't be so [adj]".


At last, I want to make a clarification that, when I searched "什么来着" as keyword with Google, on the first page, I saw 10 results, each of which contains the complete "什么来着";however,when I searched "什么来的”, I saw 11 results on the first page, only 4 of which contains the complete "什么来的“. Also, you could see "什么来的" as in "他是冲什么来的？(what does he come for?)”,but that's another story.