Which local download version of CiviCRM is most like the Spark and what aspects of Spark can be customized?
Spark is identical to the version of CiviCRM you would download, except that a) you can't install new extensions (it comes with iATS, PayPal, Shoreditch and Mosaico if I recall).  You're also limited in the number of contacts you can have.

Note that a "local" copy is a bit misleading in that you need to run a webserver to use CiviCRM.  While you could theoretically do this locally, it's requires tech chops - and if you have the capability to set up a local server, you probably don't need Spark.

Spark supports embedding basic web page forms in a separate CMS - but those forms are quite basic and can't take payments.  Payment forms will exist in the Spark install.  When discussing "CMS Integration" with CiviCRM, you're generally discussing options that are only available when installing CiviCRM within the CMS you're integrating with.