Could this be used to escape from a Demiplane?

No, as a Demiplane is a pocket dimension with nothing beyond its confines.  There’s no distance you can travel, or hole deep enough to get out of a Demiplane.  There’s no connection with any other Demiplane either as each area created in the Demiplane is it’s own dimension. Kind of a dark thing to think about if you get trapped there... 

Perhaps I should give context. Could one use Dimension Door to escape if the door for the Demiplane had not been closed yet? As if one were standing in the "room" and could see the material plane beyond the doorway. Could it be used to move ones self to a point you can see on the other side of the door?

As per Demiplane: “When opened, the door leads to a demiplane”

As a DM, I’d rule that the Demiplane connection only exists when the door is open, not when it is closed.  So DD would work through an opened Demiplane door, but not a closed one. 

if you can see the other side in the demiplane you can leave it... why would you waste a spell to get to the other side? unless you are blocked by something else...

I would rule that you could teleport as long as you can see it and it nothing blocks teleportation. 

My interpretation of Demiplane is that you can only see out of it while the door is open. after the door shuts you cant see out of it unless you use another spell to see out of it.

Does the door stay there for any period of time?

Such as go through the door, collect and item, then go back through the door once mroe to my original position?

Quote from Heroic_Dad_Mode >>

Does the door stay there for any period of time?

 

Such as go through the door, collect and item, then go back through the door once mroe to my original position?

The spell duration is “Instantaneous” so unfortunately (unless your DM chooses to make an exception) no. The door opens, you (and potentially a friend) pass through, and it closes. If you wanted to return you would have to wait until your next turn, and cast it a second time. 

Does this spell maintain momentum? If my character was falling, could he dimension door  to a safe location without feeling the "fall"?

In general, do teleportation-tagged spells maintain momentum?

Could you use this spell on 2 others, not yourself, and teleport them away?

Quote from A_Lommy >>

Could you use this spell on 2 others, not yourself, and teleport them away?

 No, the text specifies you teleport yourself.

Could a simic hybrid with the manta glide adaptation cast dimension door to teleport themself 500 feet into the air, then use feather fall as a reaction to slow their descent while they glide away?

Quote from WilyMethos >>

Could a simic hybrid with the manta glide adaptation cast dimension door to teleport themself 500 feet into the air, then use feather fall as a reaction to slow their descent while they glide away?

 The manta glide doesn't act like feather fall. You subtract 100 ft from the distance you fall when calculating fall damage so you'd still take 400 ft of damage. You could teleport 100 ft into the air and glide down safely.

Do you have to be touching the willing creature to teleport with them? It just seems to say that they must be within 5 feet... 

Would this count as a planar portal?

Can this spell be used at a higher level?

IE, i have no more 4th lvl slots, but i do have a 5th lvl. could i use that to cast it?

Quote from Stormfury >>

Can this spell be used at a higher level?

IE, i have no more 4th lvl slots, but i do have a 5th lvl. could i use that to cast it?

 
Yes, but with no additional effect; all spells can be cast using a higher level spell slot.

Quote from Delostacia >>

Do you have to be touching the willing creature to teleport with them? It just seems to say that they must be within 5 feet... 

 
No; it just has to be your size (or smaller), willing and within 5ft of you.

I think that if say, you where a gnome casting this spell, and you are using it to teleport one of your larger companions, that character could go prone and crawl through it 

If someone is willing but has a condition that would prevent movement but still aware of surroundings such as grappled, restrained, incapacitated, etc., could you still take them with you with Dimension Door as long as they're within 5 feet? Like an ally trapped in a giant crocodile's mouth.

Yes, there is no actual need to go through a "door" since it states as a teleportation in the first line

Can you use the spell as trap spell (ex place in front of enemy and it teleport above them and drop them on the ground or a pit)