To answer your two questions:
Can she travel from the US to India via Doha, Qatar with passport validity of only 50 days?
Yes. The passport is valid for entry into India till the day its valid (the midnight of it, to be precise); and you are arriving well before its expiry.
Transit in Doha does not require a visa. If you have an extremely long transit, the airline will arrange for accommodations and any associated visas.
Can we make a reservation on Qatar Airways for the roundtrip travel
using her current passport and before the return from India to the US
update passport information on the reservation?
Yes. If you have visas in the old passport that are still valid, it will be returned to you upon renewal. You should carry both passports with you when you travel out of India; until such time that [a] you have transferred the visas from the old to the new passport or [b] the visa in the old passport expires.
Your eligibility to enter the US will be confirmed when you arrive for your return flight; you will need the old and the new passport to confirm this.
All travel stamps will be done on your new passport (even if your old passport has empty pages).
Your API (Advanced Passenger Information) will be updated when you check-in and your new passport will be used, and your old passport verified for visas.