To install task server on any server operating system, it is required to have the following components. If the requirements mentioned below are not met, then task server will not function correctly:

Prerequisites: 

Check the KB 164960 for the complete required server roles and features.

1) IIS is Required 
- Select "Web Management Tools" with all sub-components (IIS 6 Management Compatibility, IIS Management Console, IIS Management Scripting and Tools, IIS Management Service)
- Select "World Wide Web Services"
- Select "Microsoft .NET Framework 3.0" (If the option is available)
- Make sure that ASP.NET and ASP are selected

2) Application Server is required
- Select "Microsoft .NET Framework 3.5" (If the option is available)
- Select "IIS Support"
- If Site server is also going to be used as Mobile Management server, then enable "Message queuing Activation"

3) Firewall (Windows or any other third-party) needs to have provision for inbound/outbound connection allowed on Port 50120, 50121, 50123, 50124. Please note this ports are specifically used for Task server, Symantec Management Platform uses other ports as well for various purposes. For more information, please refer to product user guide.

4) Microsoft .NET Framework 4.5.1 (or higher) is also required. 

While adding the Task Service in the SMP Console, it will also show you what we are expecting to have setup (if you right-click on Task Service>View Prerequisites for Site Service):

Post Configuration:

Task Server installs its own Application Pool into IIS7. This pool is configured by the Task Service to have 'Integrated' Managed Pipeline Mode.  Every time Task Service restarts it makes sure the settings of the application pool are correct, if not, the settings are corrected and Managed Pipeline Mode is set to 'Integrated' back.

Question:
Pre-SMP 7.5 SP1, can I call the application pool name: 'TaskServer' for example.  I could then assign the Client Task Server to that application pool with .NET Framework 2.0, Managed Pipeline Mode (Classic or Integrated), and the Identity of ApplicationPoolIdentity. Doing that should allow the product to function just fine.  Correct?

Answer:
Correct. But from ITMS 7.5 SP1 Task Server creates its own application pool. That is introduced in order to have isolated application pool (exclusively for Task Server) what can be configured independently of other pools (what can be used by other applications).
So, You are correct, name of pool does not actually matter, properties of pool do matter. However,  starting from 7.5 SP1 we would recommend to use Task Server’s application pool.