There is no context for these sentences:
(1) ある男と自己紹介するために、夜明けに家を出かけた。
(2) どっかの男と自己紹介するために、夜明けに家を出かけた。
both best translate to:
I left my house at daybreak to introduce myself to a man.
#2 explicitly states (I did not know where the man was from)?
#1 tacitly hints (I did not know where the man was from)?
Are those translations correct?
#1 does not sound like it would be used in a daily conversation among friends?
#2 is too informal to use in business Japanese?
First, "to introduce myself to X" is "Xに自己紹介する", not "Xと自己紹介する". "家を出かけた" is a bit strange, too, and you should say either "家を出た" or "出かけた" (without 家を).
ある男に自己紹介するために…
This is the natural choice, because you probably know who you're going to meet, and at least his name. ある男 here means "a (certain) man". This phrase is fine in a novel, but if this is used in a business setting, you should replace this 男 with 男性, 男の人, or simply, 人.
In casual conversations, people usually say "ある(男の)人に自己紹介しに..." or even simply "ある人に会いに..." because するために sounds stiff.
どっかの男に自己紹介するために…
This is highly unnatural, because it sounds like you're going to introduce yourself to a random person out there. Or it would sound as if you were thinking the person you are going to meet were trivial, and his name were not worth memorizing.
And どっか is too casual in business settings, too.
