I have to install the Service Pack 3 on an SQL instance running with the SP1. No test, the install is directly on the production server. Well this is not a complex task and the topic has been already ask for other version, but I'd like to be sure to not forget some steps and having feed back from experienced dba is always welcome.
Windows Server 2008 R2 Service Pack 2
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. Service Pack 3 for SQL 2008R2. All our SQL 2008R2 Servers are at the latest patch level of 2008R2. We had installed SP3 first on Development, Test an Acceptation. Wait 2 weeks and dit some application test before we installed SP3 on our Production Environment. Before we installed SP3 on our 2008R2 Servers we stopped our maintenance jobs and SQL Server Jobs which had connection with some important applications. We also stopped some web application which had connection to our SQL Server databases.
Server 2008 R2 Sp1
If the SQL Servers are VM's you could make a snapshot. We only make a Full Backup and stopped some important web applications. Then we installed SP3. You don't have to stop SQL server agent due the installation process, SQL Agent and SQL Server will be stopped and started. Be aware sometime there is a pending reboot on the SQL Server.
Server 2008 R2 Sp1 64-bit Download
During the installation proccess at the end you could receive an error with pending reboot. You first have to reboot your SQL Server then you can install SP3. After SP3 is installed you should reboot your SQL Server.