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.
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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.