Is it possible to restore a database with database backups starting on top of a restored server point in time snapshot that is out of date?
can I use the database backups that have occurred (on the original server) after that full server snapshot was generated, to bring the newly restored server up to date?


No.  You cannot manually switch an ONLINE database to RESTORING and apply log backups, and you cannot attach a database and leave it in the RESTORING state.  

This would be a useful feature, and it's been requested before.  See eg here.  


  Which kinds of database backups would I need available to be able to do so?


Any full backup, optionally a differential backup, and then the log backups from that point forward.  Note that it doesn't have to be the latest full backup, as long as you have all the log backups going back to that full.  Full and  differential backups never break the log chain.