"Your bed is near the door."
Translation:Твоя кровать около двери.
What is the difference between all these 'near' words? Около, возле, у
"Около and возле are equivalent prepositions that require the genitive case and are close to the English preposition 'near' - "около здания" means "near the building", and so does "возле здания".
At least, that is what dictionaries tell you; from personal experience, when describing city planning около has a connotation of a more broad vicinity than возле. It may have to do with the etymology: Proto-Slavic *kolo ("circle") + the prefix o- ("around") would imply "somewhere around that place".
Близко is very similar in meaning and in syntax requirements to the English 'close' - it is an adverb, and as such it requires another preposition, like от. However, to me "близко от здания" sounds a little off, so I'd use the equivalent synonym недалеко in this context - "недалеко от здания".
Близко along with недалеко and рядом are used in standalone sentences like "Где театр? - Он близко/недалеко/рядом." ("Where is the theatre? - It's close/not far from here/nearby.")
Около and возле are also adverbs as well as prepositions, and can be used in a standalone sentence ("Я живу тут, возле" - "I live here, nearby"), however it may sound a little strange, especially with около."
You can't. The usual patterns fail with words ending with "-ь", so you just need to memorise those. Fortunately, both words for bed, "кровать" & "постель" are feminine, so at least you do not need to memorise any distinctions. (While often - but not always - interchangeable, кровать means a bed as a piece of furniture while постель is a bed you lie in. In this example кровать is a much better choice, but you'll be understood regardless.)
Whenever I'm not sure about the gender of a word ending in -ь, I just assume it's female because most of them are (there are three times as many feminine as masculine words ending in a soft sign). However, some common masculine words include the ones ending in -тель, words that refer to men (for example король, king), the names of the months and the word день (which also in Latin and Spanish looks feminine, but is masculine).