Your original question asked two questions, so I have had to edit it down to one. In this poem it is important to realise that Death is personified as a carriage driver who politely stops to collect the speaker. It is interesting to see how Dickinson uses this metaphor to "tame" or "domesticate" the most awesome and inevitable of human experiences - death. The literal elements of this metaphor are simple: dying is compared to an unexpected ride in a horse-drawn carriage.
Therefore, the speaker spends most of her time sat next to Death in this carriage. Note how the first stanza describes it:
The Carriage held but just Ourselves -
This is part of the unforgettable nature of this poem - it makes us see death as nothing terrifying, but just a normal carriage ride with a polite gentleman who kindly picks us up at the right time.