Learn Russian: Правая дверь новая.

In Russian, the present tense of the verb "to be" (am, is, are) is usually skipped. So, Правая дверь новая literally translates to "Right door new".

How can I tell the gender of the word дверь (door) since it ends in a soft sign?

Nouns ending in a soft sign (-ь) can be either masculine or feminine, which you just have to memorize. However, looking at this specific sentence, the adjectives правая (right) and новая (new) both have the feminine ending -ая. This is a huge clue that дверь is a feminine noun.

Why do both правая and новая end in -ая?

In Russian, adjectives must match the gender of the noun they describe. Because дверь (door) is a feminine word, both "right" and "new" take the standard feminine singular ending -ая instead of the masculine ending -ый.

How does the word order make this a full sentence instead of just "the right new door"?

If you wanted to say "the right new door" as a single phrase, you would typically group the adjectives together before the noun: Правая новая дверь. By placing новая after the noun here, it acts as the main point being made about the subject, giving us the complete sentence "The right door is new."