SSMS 10.5 alongside 17.7
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.