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It is important to understand that there is a third piece of software involved here. Bitbucket is not a Source Control Management system like Subversion, it is a management layer for an SCM - underneath Bitbucket, there is a git system running.
What you would be migrating between is Subversion and git, not Subversion to Bitbucket. You've then chosen to use Bitbucket on top of git, to manage it all.
Atlassian have a really good guide to doing this - see https://www.atlassian.com/git/tutorials/migrating-overview
Note that although git will import every commit you have made, with a proper history including the core stuff like who did what and when, it does work very differently to Subversion, so your versions may not work the way you expect from Subversion.
The second part of your question is about build services, not source control. Importing packages or databases, or compiling code is a build process. You will need to look at what your current build processes do (note that they are not in Subversion, they just read the code from Subversion when building), and decide whether you want to rebuild them in Bitbucket-pipelines (if you're on Atlassian Cloud) or Bitbucket-runners (if you're on Server/DC), or keep whatever is currently doing them, and tell it to pull the code from git instead of Subversion.
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