"This is my door."
Translation:Das ist meine Tür.
Dieses is used in front of a neuter noun: Dieses Haus ist schön = "This house is nice"
Diese is used in front of a feminine or plural noun: Diese Tür ist geschlossen = "This door is locked"
If you don't have a noun directly after it, then you use dies: Dies sind meine Kinder = "These are my children"
Tür is feminine, but dieses is for neuter nouns. It doesn't fit together.
Better is Dies ist meine Tür. That's like saying "This is my door", where "this" doesn't have any 'gender information' on it at all. If you say Diese ist meine Tür you're more strongly giving information to "this", making it more like "This door is my door" or at least "This one is my door".
No, "dies" and "das" mean "this" and "that" respectively, although I think the two words are more interchangeable in German than in English. It just so happens that "das" also means "the" for neuter nouns (in the nominative & accusative cases). "Der/die/das" can also be used to mean "which/who", though: "Hier ist ein Schloss, das alt ist." = There's a castle here which is old. "Ich esse einen Apfel, der gesund ist." = I'm eating an apple, which is healthy. "Ich habe eine Schwester, die laut ist" = I have a sister who's loud.