Since you have already installed Client Tools on the Passive node, you need not do it again.

I am not aware of any such tools.The requirement varies case by case

Is thidb the default database for the account which is used in linked server? Also does the login has access to that database?

A reboot is required after applying SP3. You need not stop any SQL related services before applying the Sp3.

You need to install the SP3 on the passive node in...

The database backup cannot be taken amidst a transaction, like in trigger.

Can you please post the SSIS package and the job script?

Did you try changing the service account of "Performance Logs and Alerts" service? May be you can try starting it with the SQL Server service account.

Please post the Screen shot or the script of the job you are trying to execute. Also let us know while executing which task you get this error.

What is the problem and the related error messages?

Please do it for the system databases except tempdb !

osql /E /SSTCSQL /dDB_PROD /h-1 /b /Q"set nocount on select getdate(), 'F4111 ',count(*) from [PDDTA].[F4111] WHERE (ILMCU LIKE '% 242%') OR (ILMCU LIKE '% 241%') OR (ILMCU LIKE '% 232%')"...

No problem. Yes please go ahead with it

Check Database Integrity (should I include indexes?)

Please include indexes

Shrink Database(do I need to do it?, if yes I...

You may want to consider re-indexing table by table. Between two re-indexes you can also give a wait time of a minute or so. Also you can fire a transaction...

I second that. sys.servers should give you the details.