The most obvious verb to use here is venari, which you've found. The difficulty is, that it's what's called a deponent verb – it has, for the most part, only passive forms; and these are, for the most part, only active in meaning. So it isn't possible to use both active and passive forms of the verb, as is done in the English sentence that you're translating ...
There is, also "carpe iter" = "take one's (your) way" (Pock. Ox. Lat. Dict.). This "iter" is a neuter noun, therefore, the accusative is the same as the nominative. Definitions include: journey; march; route; road; foot-way.
Close! You'll want Carpe Viam, with an M at the end.
Via isn't a perfect translation of "lane", and carpe doesn't actually mean "seize" (it's closer to "harvest"), but the saying is well-enough-known in English and the meanings close enough that it'll be easily understandable, and the pun works better with viam than with any more-precise word.
This is kind of a subjective question. Here's what I can think of:
Vita et Mors (or Mors et Vita) -- I like this for its simplicity, readability, and recognizability.
Vita Morsque (Or Mors Vitaque) -- It means the same thing "Life and Death" but it is more idiomatic Latin, I think.
Res Vitae et Mortis -- "Res" means "matters, goings-on." This means "...
The origin of the town's name is ultimately unclear, and is missing from many dictionaries. The -don is probably a coincidence, though, and we can probably assume it's Pre-Greek or some borrowing from nearby tribes.
Similar words that might give a clue (καλύβη "kalubē", καλυδίλα "kaludila") are classified as Pre-Greek by Beekes (p. 628 for a smattering of ...