Want to know that Robot manager is inbuilt with pega Version 8.1.3?

Robot Manager doesn’t come built-in with any version of pega platform version. You need to follow the version documentation to find support Robot Manager version, steps for importing and configuring available here : https://community.pega.com/marketplace/components/pega-robot-manager

What is the recommended pega robotics studio version to install for pega 8.1.3?

Pega Robotics version is not dependent on Pega Platform. The minimum version of Robotics version for a Robot Manager is listed in each of the version release documentation. For version 8.1.3, only Robot Manager v6.1 is compatible and requires 8.0.2025 and later.

What is the difference btw Pega Robotics version 8.0.2025 vs 8.0.2032 Vs 19.0(which is very latest)?

There has been BUG fixes and enhacnments to our product between v8.0.2025 and v8.0.2032. More details available here : https://community.pega.com/knowledgebase/cumulative-build-notes-pega-robotic-automation-80-sp1

You can check 19.1 initial build release documentation get details on what has been modified between the 8.0 SP1 and 19.1 release here: https://community.pega.com/knowledgebase/documents/pega-robotic-automation-version-191-release-notes

What will be the supported IE version for robotics(8.0 and latest)?

All the supported IE release versions are listed in Pega Robotics Automation Application Support Matrix

I hope this answers all of your queries.