1. I've a specific set of 20 Business Users. What should be my approach.
2. Is it possible to have a personal gateway installed on end user desktops or only On-Premises gateway is an option?
The refresh is handled by the owner of the report. So if you are sharing it with 20 users, you would need to have the gateway, not them. I would recommend you use the On-Premises Data Gateway / installed on the same machine as the cube.
3. Is Azure Active Directory is a compultion to connect with Windows Active Directory or is there any workaround wherein I can avoid AD integration. I learnt about Mapping Users where I can map Power BI email logins to windows Logins (for SSAS cubes), keeping in view that I've only 20 business users and I am OK manually mapping the users?
You need Azure AD otherwise the users can't log in. You can manually create the users in AAD, you just need to make sure their UPN matches the on premises AD. Obviously, any changes to passwords or such on the AD side will cause issues if you don't sync.
Mapping users only applies to granting them access to the tabular model. If they need accounts to access the dashboards you share, then they need their own user account in Azure AD.
4. Do I need Pro license of is there any approach in which I can manage with Free lIcenses?
SSAS is an on premises data source, it requires a Pro license. It also requires that all end users have a Pro license.
5. When we publish a report and share the URL , how can it be exported to excel?
I believe Analyze in Excel just became available for use against SSAS sources. Otherwise each visual has the optiont to export data. (Both of these can be disabled via the Admin Portal -> Tenant Settings