I found “a 3 meters long door” is correct. I also want to know if “a 3 meters lengthened door (3m+3m=6m).”
"3 metres long door" is odd, but correct. Normally you would talk about the height of a door (a standard door is about 2m high, so a 3 meter high door would be surprising, perhaps on a castle?)
It looks very strange to talk about a door being "lengthened". Doors are made of wood and metal (etc) and you can't easily stretch them!
I suppose you could say
The carpenter lengthened the door by three metres.
But the situations in which this would be used are very exceptional.
When talking about things that can be lengthened (such as cables, which can be lengthened by connecting an extension cable), you would say
A cable (that was) lengthened by three meters
And a 3m cable lengthened by 3m would be 6m long.
"A 3 meters long door" is incorrect. Correct grammar would be, "a 3 metre long door". When a unit of measure (metres, hours, kilograms, ...) is used as an attributive adjective (an adjective that comes before the noun), it is not pluralized.
As for "a 3 meters lengthened door", the grammar is iffy, and it's not natural. Better would be, "a door lengthened by 3 meters".
Also, assuming you're talking about a normal physical door, we talk of doors in terms of their width or height, not length. The verb for increasing width is "widen". There is no verb for increasing height, so just "extend".