SSMS 10.5 alongside 17.7 To set the scene, I maintain 4 local SQL Server 2008 R2 instances. 
At some point in the near future, 3 of these will be upgraded to SQL Server 2017 and migrated to azure.

I am currently using SSMS and SSIS 10.5 to manage these servers and develop ETLs.

Can SSMS 10.5 and 17.7 be run side-by-side, and can I do this before we've even upgraded any of the instances to 2017? Is there any benefit of doing so?
I'm considering forking out for ApexSQL but some of the tools aren't compatible with 10.5.

Can SSDT also be installed side-by-side with SSIS 10.5?

Thanks.
You can use the latest version of SSMS with SQL 2008 R2. Im not aware if any backward compatibility problems. 

It will also work with SQL 2017 and Azure.

SSDT is a tool used to develop SSAS, SSIS and SSRS Reports. It wont cause any problems installing alongside SSMS.

SSIS changes a far bit in 2012 namely in the use of SSIS catalog. IMO its a great improvement. Depending upon your layout it may be feasible to upgrade SSIS first or independantly.

Note that once you open an old solution in a new version of SSDT or visual studio it will want to upgrade the solution. Once that happens you wont be able to open it again with an old version. Ive been through a similar scenario as you describe. In the end i found it was just easier to take the plunge and upgrade to the current client tools.