Yes, you gain an additional attack

From the FAQ:

Does the extra attack from spending ki as part of a flurry of blows stack with the extra attack from haste?

Yes. The extra attack described in the ki pool ability doesn't say it works like haste, nor does it say that it doesn't stack with haste, so the monk would get two additional attacks (one from spending a ki point as part of a flurry, one from haste).

It stacks

Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.

If you take the attack action, you attack twice.

At the start of your first turn of each combat, you walking speed increases by 10 feet, which lasts until the end of that turn. If you take the Attack action on that turn, you can make one additional weapon attack as part of that action. If that attack hits, the target takes an extra 1d8 damage of that weapon's damage type.

If you take the attack action, you can make one extra attack.

Nothing in these two statements prevents the other. One changes your number of attacks to a flat number (2), the other says "whatever it was before, +1".

2+1 = 3, you get three attacks in the first round of combat, and one of those 3 attacks deals extra damage. You get only 2 attacks for the rest of the combat.

On multiattack: Multiattack is a monster specific action. It is NOT the 'Attack action', it's a specific action some monsters have, that determines what it can do with those attacks. This is to prevent, for example, a bandit captain from using its multiattack to make three grapple attempts in a single round. Both your features require the 'Attack action'.