Partial answer (first part) -
--Edited in an attempt to further clarify my thought process--
Each line has two parts: a word and a clue. We're grabbing the common characters between the two
Tomato and the mouse’s nemesis
"the mouses's nemesis" - Jerry (the antagonistic mouse) considers Tom (the cat) his nemesis
The shared characters are TOM
Nor is it a connective option that
"a connective option" - or is a conjunction that has common letters with 'Nor'
The shared characters are OR
Crows go gently down the stream.
"gently down the stream" - I think of row, row, rowing a boat
The shared characters are ROW
Put them all together and you get TOMORROW
Tomorrow is the next day and therefore begins the future
'Tomorrow' never actually comes, it is always 'Today'