With Extra Attack, if you make *one* attack with it as bonus action, will it be 2 attacks in that bonus action of still one?, because an attack bonus action is still an attack action turned bonus, so will you be able to attack twice with each an extra attack?

etra attack only i say activate if its the attack action the wepon clearly stats that its one attck as a bounus action

No, Extra Attack is nested within the main Action category and you can't swap em around.  No method exists for getting an extra bonus action, either.

Would the bonus attack have the full damage bonus?

Thirsting balde with scimitar of speed a and lifedrinker... 

atack action gets two attacks because of thirsting blade and then the third bonus action attack all getting the necrotic and slashing and proficiency bonus if proficient with scimitars. 

Quote from RipleyRiley >>

Would the bonus attack have the full damage bonus?

Yes as it's still a main hand attack, not an offhand attack as 2 weapon fighting so all bonuses such as proficiency bonus(if you are proficient in scimitars) would be included in the attack. Eligible for bonuses such as divine smite as well.

Would also add you could use your action to do something else and still attack once with the weapon. As it doesn't state you have to take the attack action to get a bonus action attack. Simply that you can take one attack as a bonus action on each of your turns. So you could cast a action spell and still bonus action attack once, or dodge action and attack once as bonus, etc.

I'd say yes because it's still in your main hand, unlike in two weapon fighting where you don't get the modifiers with your off hand. I think that's what sets this weapon at very rare rather than just rare, you get a full extra attack as a bonus action. Sorry to dig up this comment... Hehe

Ability modifier applies to that damage, not proficiency.

this can be great on a rogue as they may attack with this weapon as a bonus action without their action being tied up. If it lands, they can use their action to take the dodge action or even ready an action to attack on another creatures turn for a chance to get sneak attack damage a second time in a round.

Or you can just have two +2 scimitars.

 but that also takes up both hands, scimitar of speed is good on a sword and shield with dueling. And it'd gunk up any item interactions you wanted to take. AND its expensive/rare, and depending on the party comp it could be item hogging (gives me an idea for magic weapon that turns into 2 weapons). It'd also be weaker without dual wielding fighting style. 

Not needing two weapons for a bonus attack is the scimitar of speeds whole gimmick. 

It says "you can make one attack with it as a bonus action on each of your turns", not taking the Attack action as a bonus action but making one attack as a bonus action.

So if you take the "Attack Action" with a class that has an extra attack such as a level 5 fighter.  That level 5 fighter would be able to attack with this scimitar 3 times.  (2 as part of their action and 1 as part of their bonus action)

In the hands of a bladesinger using his blade song this would be ugly! 

Scimitar of Speed + Haste = Win

bladesinger's best friend (use it with spirit shroud)

Can dis be a dou’b skimmitar?

Ask your DM. I'd allow it at my table.

Arcane trickster rogue has 2x bonus action

No. It's the 2 attacks from the action and one attack on the bonus action, so you get three.