Do the skill increases during the tutorial count towards attribute bonuses,or do they only start to count towards attribute bonuses after you exit the sewers ?

They count towards attribute bonuses.

They count only if they're "permanent" skill increases.

For instance if you get 4 raises to Blade before choosing your class, and then make Blade a major skill so that there's now no raises to it you will get no credit towards STR increases. However if Blade is still a minor skill you now have 4 points toward your STR attribute at level up. This is a good argument for using a melee skill that will end up as a minor skill before you choose your class.

Mara

Yes. The skill increase at the moment you leave the sewers count.

You can juggle the skills when selecting your class or just before leaving. But it's the skill increases just as you leave the sewers that count.

To expand on this a little, as your character goes through the tutorial, the character gets experience points for each skill. That is kept track of behind the scenes (you never see it without a console command). But the amount of experience you need to advance a skill depends on (1) that skills level - the higher the skill the more experience needed, and (2) whether the skill is a major and specialized - majors and specialized advance faster. So the number of skill advances you get is affected by your characters choice just before leaving.