I need some basic information on SAP: What database platforms are supported? What operating systems are supported? And what kind of support is there for a mobile environment?

Have a look at http://service.sap.com/platforms*. All database and operating system platforms supported by SAP are listed there.

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How many backups are required from preparation to end of an upgrade and at which phases?

Take your daily online backups during the uptime phases as you would normally do. Please note that your system is still being used productively and you cannot afford data loss. In case of disaster, you will restore the daily backup, roll-forward up to the crash and roll-back the upgrade to re-start it from scratch. A backup is definitely needed at the beginning of the downtime phases as it is the point of no return. Another backup is needed before the go-live.

How can I clone an SAP system for a sandbox purpose? For example, our Enterprise production environment runs on 4 application servers and an Oracle database all on AIX OS. I can restore a copy of the database from a backup and copy the /usr and /sapmnt directories from one of the app server to the target sandbox, but what files should be edited/deleted to have the new system run on its own?

Congratulations -- restoring your database and getting it up and running is the hard part.

The rest should be easy.

The following needs to be changed after restoring the SAP file systems:
• The instance, default and start profiles (they reside in the profile directory).
• You need to copy the environment files from the adm homedirectory from your source system. The environment variables values should be changed accordingly.
• Copy and modify the database configuration files (init.ora, init.dba … )
• Don't forget to set the port names and numbers in the services file

What is the best way to create indexes in SAP? I want to avoid problems such as the index not being created in the database level. Should I first create the transport for an index and then create it in the database level?

Customer objects should always be created on the development system and transported to the other systems in your SAP landscape.

Our development system is way out of sync, both configuration- and data-wise. We would like to refresh our development system from the production system. How can this be accomplished? What things we need to be aware of? I know we can restore the production system (PRD) from from tape to the development system (DEV,) but that will wipe out everything including transports, notes history, etc

The system copy guides can be found at http://service.sap.com/instguides*. Different backup/restore procedures exist for the different databases supported by SAP.

If you restore PRD over DEV, it will wipe out everything. You should therefore find a point in time in which development projects finish and are brought into production. You should only re-create DEV from a backup of PRD if both systems are more or less in sync or if your development teams agree that certain developments are lost.
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Is there a way to determine the amount of data in a client for a given company code (client size by company code)?

Yes there is -- have a look at OSS NOTE 489690: CC INFO, copying large production clients. You can also determine the size and the storage requirement of your clients with reports RSTABLESIZE or RSSPACECHECK (see also note 118823).