ZEDs require one free checked bag, but LH may allow a second, especially if they do for revenue passengers. I'd take a look at the policy for paying customers and expect that (but if you can do one bag, I would).
If you're staying on the same airline, and on a single ticket as you say, they'll probably be able to do it all the way through. Kind of depends on the airline, but most don't have a problem with accepting transfers.
What's your nationality? I can check TIMATIC when I'm at work to see what exact requirements are. You say you have a valid US visa, are you a US resident, even if a citizen of another country?
UA has 10/56 available seats in C/Y to DEL that day, 14/52 to BOM. I've actually been watching those loads religiously for a friend/forum member the day before. Those have maybe a dozen more seats per flight, but I'm giving him decent odds with the numbers I've been seeing. Still, check again a few days prior.
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