Ytdyklly

Assuming that a Mordenkainen's Magnificent Mansion is conjured and the entrance is closed, could a creature make use of the Knock spell to open the door to the mansion?

Would the gateway be not considered as an object that Knock can interact with?

Would the creatures get immediately expelled even if the entrance is open (since they were not designated to be allowed inside the mansion at the time of casting)?

Knock cannot open the portal to a Magnificent Mansion.

From Magnificent Mansion:

You conjure an extradimensional dwelling in range that lasts for the duration. You choose where its one entrance is located. The entrance shimmers faintly and is 5 feet wide and 10 feet tall. You and any creature you designate when you cast the spell can enter the extradimensional dwelling as long as the portal remains open. You can open or close the portal if you are within 30 feet of it. While closed, the portal is invisible.

Let's first assume that you overcome the fact that the Mansion's portal is invisible with a See Invisibility spell (which is reasonable, since Knock and See Invisibility are of the same level).

Now, let's take a look at Knock:

Choose an object that you can see within range. The object can be a door, a box, a chest, a set of manacles, a padlock, or another object that contains a mundane or magical means that prevents access.
A target that is held shut by a mundane lock or that is stuck or barred becomes unlocked, unstuck, or unbarred. If the object has multiple locks, only one of them is unlocked.
If you choose a target that is held shut with arcane lock, that spell is suppressed for 10 minutes, during which time the target can be opened and shut normally.
When you cast the spell, a loud knock, audible from as far away as 300 feet, emanates from the target object.

Emphasis mine. Note that the effect of the Knock spell is not to open a door, but to unlock, unstick, or unbar it, allowing it to be opened normally. If you cast Knock on a locked door, the door unlocks but remains closed. If you cast Knock on a closed, unlocked door, the spell has no effect. In either case, you still need to reach out, turn the knob, and physically open the door.

The portal to the mansion is not barred, stuck, or locked with a mundane lock. Neither is it locked with an Arcane Lock spell. The Knock spell effects doors that are sealed by these means, and only these means. The portal to the Magnificent Mansion is not locked, barred, stuck, or sealed with an Arcane Lock. It is simply closed. Therefore, Knock will have no effect on it, just as Knock has no effect when cast on a door that is closed but unlocked.

No. I'm basing this on Knock's requirement:

Choose an object that you can see within range.

and from Magnificent Mansion:

While closed, the portal is invisible.

Since you cannot see the entrance to the dwelling, it is not a valid target for the Knock spell.

Based on the wording of knock:

Choose an object that you can see within range. The object can be a door, a box, a chest, a set of manacles, a padlock, or another object that contains a mundane or magical means that prevents access.

you could have Knock reveal the invisible portal to the mansion as that is something which is opened/closed. However,knock lists things which require some sort of key or keyword and as access to the Mansion doesn't I would compare to Magic Circle with how the effect is looking as it were at who is trying to enter and stops them.

In the case with Magic Circle I would not allow Knock to be used to allow an otherwise blocked creature from entering and similarly I also would not allow Knock to give a creature access to the Mansion.