What is the procedure for doing an MBA while working at TCS, or taking leave without pay to pursue an MBA? Also, does TCS offer any sponsore

If you're doing an online MBA do it in parallel , no one can question you for this

I think both possible .. TCS has some tieups to some universities.. check with ur HR

Check with your project TD spoc they help your MBA and your work performance also will be considerd for MBA grading it seems last i heard.

Attempt CAT, goto interviews, once you get the results, apply for LWP citing Higher Education. Check the policy for higher education. Your LWP will be approved before your college joining date.