Is there a need for
down time while tool is running?
If not, should we
disable analytics during the process?
Thinking the site should be live but I do not want to have new data going into
Mongo that will not be captured by the tool.
Data migration is an asynchronous process and you don't need any down time, also It is recommended to install xDB Data Migration Tool in Content Management server. So I guess during the migration you can use your Sitecore server like how you use regularly, I mean analytics can be enabled while running data migration.
From Sitecore Document
Depending on the amount of data you have in MongoDB, it may take hours - or even days - to migrate all of your data. The migration process runs asynchronously on the Sitecore server, so you can continue to use your Sitecore server during the migration process.