Why the abandoned carts are not getting synced with my mailchimp account?

The carts are starts syncing only after the first products, orders and customer synchronisation is finished; this appears as "Account synced since: xx-xx-xxxx". You would also make sure that you had enabled the Abandoned cart section in the bottom of the mailchimp's configuration page in your magento.

My magento only sends abandoned carts of registered customers, is that an expected behaviour?

Yes it is, anyways you can enable the option "Save Email to the Quote before place Order" to force magento to add the email address to quotes of no registered customers and then those carts should get synced with mailchimp too.

Some of my orders/quotes are not reaching Mailchimp.

Mailchimp does not support products of the types Grouped, Bundle or Giftcards; if an order/quote doesn't have at least one supported product, the order would never get synced.

Mailchimp's JS is failing to load. (See attached image)

This could be solved by deleting the 'core_config_data' row with path 'mailchimp/general/mailchimpjsurl' and cleaning cache. The configuration would regenerate automatically with the right path to the file.