For MySql to work in medium trust the host must have the MySql Connector for .NET installed in the GAC. Perahps the version they have in the GAC has bugs. When I google this error "MySql.Data.MySqlClient.MySqlException: Connection unexpectedly terminated" I find a lot of results, looking at a few of them it looks like maybe a bug in particualr versions of the Connector. I've only looked at a few of the results though so maybe further looking will find more answers.
The fact that its using /Data/Sites/2 instead of Data/Sites/1 is strange and makes me think some errors happened during setup or something. I would look in the mp_Sites table and check if the SiteID is 2, then it should be ok to have /Data/Sites/2, but makes me wonder what happened to Site 1.
If you haven't entered a lot of content yet, one option is to drop all tables from the db and visit setup/Default.aspx again and it will re-create the initial tables and data and you should be back at Site 1 and using /Data/Sites/1
Nothing in your error makes me think its a lucene issue or medium trust as those typically manifest as security exceptions.
Hope it helps,