Just how long does the 'tutorial' actually last?

Solution 1:

It took me 3 or 4 missions to get out of the "controlled tutorial" mode. See spoiler below.

The tutorial mode seemed to end (everything was unlocked) when I captured and investigated my first alien.

The tutorial (as far as I know) ends after these missions:

Operation Devil's Moon (where 3 soldiers die)
First Alien Abductions (only 2 sites, blah blah blah rocket launchers destroy cover)
Council mission to extract Anna Seng (removed in Enemy Within)
The Meld tutorial (EW only, optional, can be enabled without the rest of the tutorial)
A UFO Crash Site for a Small Scout (not sure what happens if you fail the interception, or if that's even possible)

As for the practical differences of disabling the tutorial:

Disadvantages:

You'll be dropped into the game without knowledge of extremely important mechanics such as cover, construction, panic control, and flanking, and more. No, seriously, if you want your soldiers to survive, you'll need to know how cover works.
If you're playing on Classic difficulty, then you won't get the free Satellite from the Council. On Easy and Normal difficulties, you get one anyway.
The first Alien Abductions will have 3 sites instead of 2.
You won't get the achievement for doing the tutorial.

Benefits:

You'll be able to locate your base on any continent you want, instead of just Europe or North America.
You'll be able to arrange your first four Satellite Uplinks in a 2x2 square and get more adjacency bonuses.
You'll be able to do any of the initial research projects, in any order.