Need Career Advice - International student looking to enter the industry

Hi All!
I currently work in Treasury Credit Risk Management for a BB in their India office. My work involves credit due diligence on treasury investments which include securitized and non-securitized investments, and some credit impairment model development for CCAR and CECL.
My plan for the future is to enroll in a Top 15 MFE program in the United States and, hopefully, transition into the AM industry. I know AM is quite broad but my primary goal is to be on the PM track at an HF. I am a computer science graduate from a top 5 university in India and have cleared my CFA L1.

Here are my questions:
1. How difficult it is after an MFE and my current Risk background to enter into a PM track at an HF?
2. Do HFs sponsor Employment Visas? I know the current market scenario is not good due to COVID 19, but I'll be graduating sometime in 2022 at the earliest.
3. I am currently learning a lot of products in my job( CLOs, RMBS, CMBS, Auto ABS). Should I, at this point in my career, go deeper into the same asset class(Fixed Income) or be broader with my knowledge base (like learn about equity derivatives too)? I have basic knowledge of derivatives and the Black Scholes Model.

Any advice is greatly appreciated.
Thanks a lot!