The Web Server located on MSP environment needs to communicate with the index Server directly (not sure we can via gateway proxies) Most communication is done through HTTP for solr queries however basic commvault communication between the access nodes and the media agent used will be expected to reach the index node (https://documentation.commvault.com/commvault/v11_sp20/article?p=7102_1.htm). 
Does the index server require to be on the Tenant Domain?  No, As stated above HTTP calls through the webserver/index gateway will occur for browse/search/and Content indexing jobs.  However normal communication means to the index server and related nodes will be necessary also. 
Can it be located on MSP environment and still manage indexing of an Exchange Server located on Tenant network and/or on O365 ?  I’m not sure I follow your question here however if your asking if you can use an index multiple times for different environments, the answer is yes(https://documentation.commvault.com/additionalsetting/?q=showAllIndexServersForExchangeOnePass).  However be aware of limits seen on the index node(https://documentation.commvault.com/commvault/v11_sp20/article?p=28819.htm).  If your asking whether the index node can service both on-prem and/or O365 from the On tenant side then yes. **NOTE** Be aware that access nodes for any on-prem exchange including on prem only, hybrid, and/or exchange online using local active directory all need to have the access nodes on tenants domain. 

Thanks Chris for your answers.