How would I use the same component arrangement for each instance of a sheet symbol?
If you're making multiple instances of a portion of a layout, the best way (as far as I've found) is to put the replicated circuit in rooms (one copy in each room). In some circumstances you can use a repeat block on the schematic, in others it's more clear to lay out multiple similar blocks. 

On the PCB design, you can lay out the parts in one room, then when you have it perfect, copy the room layout from the master to the other rooms (Design -> Rooms -> Copy Room Formats). It's quite powerful when used in conjunction with 'channels'. 

This Altium tutorial describes the schematic steps.

This Altium tutorial describes how the PCB steps. 

You'll probably want to change the default designator scheme- it creates really long ugly designator strings. I prefer something like R1A R2A/R1B R2B for resistors R1 and R2 in channels A and B (might be sub-optimal if you needed more than 26 channels). 

You can also use snippets.