TL;DR — I would say "Log in to" or "Log into" but not "Login to"
For me, I would say option 2 is definitely 'wrong'. From Wikipedia:
In computer security, logging in (or logging on, signing in, or signing on) is the process by which an individual gains access to a computer system by identifying and authenticating themselves. The user credentials are typically some form of username and a matching password, and these credentials themselves are sometimes referred to as a login (or logon, sign-in, sign-on).
So if I see the phrase "login to" I'm thinking of "credentials" (e.g. username/password combo) [noun] [preposition], for example:
Use your login to gain access.
rather than an action:
Use your password to log in to the system
The UK's The Guardian newspaper's Style Guide backs this up somewhat:
But you could argue that given it's in a "Style Guide" shows it's a matter of preference!
Personally, I'd go with option #3 — log in to — as "log in" is a 'phrasal verb' by itself. I see the "in" as being more closely related to "log" than to "to", so it seems odd to phrase it as "into", but it's so common to combine "in" and "to" in English, that I can't really object to #2 either :)
On an appeal to authority...
Taking similarly structured 'phrasal verbs' that end in a preposition, a more common one is probably "check in" (as in, "check in to a hotel").
Going back to the Guardian, they similar distinguish between "checkin/checkout [nouns]" vs "'check in' and 'check out' [verbs]" (though they don't call out a distinction between "check in to" vs "check into").
If you look at their usage in the wild, you get 1,510 results for "check in to" and 1,090 results for "check into" — so a reasonably even split.