Upgrading from SQL Server 2008 R2 SP1 to SP3
This seems to be a pretty exhaustive list in your question. Service Packs are generally speaking safe affairs - but every so often things can go wrong.

Ideally you would run on a test instance and make sure things are good. Since you can't do that - making sure you take backups is key.

Stopping the SQL Server agent isn't necessary - but if you have many jobs and are trying to make sure a job doesn't kick off just before you start, etc - that is a good plan. Again not necessary, though.

I also like to read the notes on the CUs released after a Service Pack and look for details about potential issues. It doesn't happen all the time, but sometimes a SP ends up breaking something and that then gets fixed in a later CU.  In your case you are going with a fairly late release because Microsoft hasn't been generally releasing CUs - only security updates for that product with the current release now 3 versions higher. You can read the list of versions and builds and service packs and fixes post service packs here.