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1. Definitely. CFA doesn’t get you a job, looking for a job gets you a job. Internships are a great way to start.
2. No, I’d definitely start looking ASAP.
3. No US experience I’m afraid. If your H4 doesn’t allow you to work, you can (1) as you say find an employer willing to sponsor a visa, but will be extremely difficult to do for an internship, or (2) get less ‘official’ internships. Interning in sites such as Wall Street Oasis (or here) could also help. Although not a strict finance internship, this may help your network. (3) See if you can apply for a different visa on your own.