Restore a .bmp file of full database backup to an Oracle 12c EE instance running on RDS

In short - yes, there are standard operating procedures for doing all of this with Oracle. You should start by looking into RMAN (Recovery MANager). I have put together a high level overview of RMAN as well as an introduction to Oracle backups for SQL Server DBAs. I suggest watching both of those and then heading over to the Oracle Database Backup and Recovery User's Guide to learn more about the options available to you.

Basically, though - RMAN can perform all of these tasks for you. If you need additional tooling, RMAN + Oracle Enterprise Manager can provide you with automation, cloning, and other reporting capabilities. Oracle Enterprise Manager Database Control (dbcontrol) will also be very helpful, it will give you additional web-based GUI tools.

To get to a point where you can get to restoring from a bare bones Windows server and cold backup file is going to take significant training. You need to work with the business to define effective RPO/RTO plans, budget training for the future, and then do what you can right now to mitigate the risks before you're stuck with an empty server.

Improving the original answer slightly...

1. You should start using Oracle Recovery Manager (RMAN) to back up your databases. Depending on how you configure RMAN, you may be sending this to NFS or to a set of disks on a different SAN. The way you configure this has a lot to do with your RPO/RTO policies. If you need to send this to a cool standby server, you should look into Oracle Data Guard. Depending on the configuration, this may take the place of either log shipping or an AlwaysOn Availability Group.
2. You can clone a database using RMAN, just make sure you have everything in place and you practice it.

Not sure what you expect from partial backups, but they are only useful for read-only filegroups:

partial backups are designed for use under the simple recovery model to improve flexibility for backing up very large databases that contain one or more read-only filegroups.

Differential backups are a different story, but they are not used as you use them. They are supposed to speed to up a full restore by skipping over some log backups, not as a replacement for full backups.

Is rather unclear what you actually have. I guess you have a full backup and 130 differential backups. In such case you can restore the original full and then apply the last differential.

My suggestion is to implement a correct backup plan (full recovery mode, periodic full backup and frequent log backups). There are many examples how to do this. Start with reading Introduction to Backup and Restore Strategies in SQL Server. Go over some tutorials like the ones from SQL Server Backup Best Practices and Articles. Understand what you're doing, don't guess.