Will I be able to obtain a transit visa at terminal 3 of AUH if needed?

Courtesy of Emirates, UAE's largest airline, and its Timatic tool, the information used by airlines for passenger entry requirements, you would be able to Transit Without Visa (added emphasis mine).

Destination - France (FR)

Passport
Passport required.
Document Validity:
Passports and other documents accepted for entry must be valid for a minimum of 3 months beyond the period of intended stay.
Warning:
Passports and/or passport replacing documents issued more than 10 years prior to date of travel are not accepted.
Visa
Visa required.
Additional Information:
Valid visas in full, invalidated travel documents are accepted if accompanied by a new travel document.
Warning:
Visitors not holding return/onward tickets could be refused entry.
Schengen "C" type visas must be valid for the visited Schengen Member State. This is specified in the "valid for" field of the visa.

Transit - United Arab Emirates (AE)
Visa
Visa required.

TWOV (Transit Without Visa):
Passengers with a confirmed onward ticket for a flight to a third country within 24 hours. They must stay in the international transit area of the airport and have documents required for the next destination.

Visa is required for transit between Dubai International Airport (DXB) and Dubai Al Maktoum International (DWC).

Your carrier, Etihad Airways, the flag carrier of the of the United Arab Emirates, also shares visa information and application links on its web site. Note, however, that the Transit Visa is intended for a stopover (for up to 96 hours), i.e., enabling you to leave the airport. If that’s what you’d prefer, you would need to apply

You can contact Etihad Airways with any queries regarding visas, by calling +971 (0)56 677 0326 or by email visaonarrival@etihad.ae

In summary:

1. Can I get a transit visa at T3 of AUH (being Indian)?
2. Do I even need one if I intend to stay at the airport?
   If I get one when I reach, I will obviously go out and explore.

No, only certain nationalities are able to obtain a Visa on Arrival. However, you do not need one as you are able to Transit Without Visa, but must stay airside during your layover. If you want to go out and explore, then you would need to apply, in advance, for a Transit Visa.

3. What's the worst that could happen if one was absolutely mandatory and I don't have one (can't get one now obviously so close to the travel date)? Can I pay a fine and board my onward flight? Can I be stranded? Can I at least book a flight back home from the airport?

With your France Schengen visa, you will be allowed to board in Delhi; if a visa were required, boarding refusal would occur before travel even begins, in Dehli. You wouldn't get as far as Abu Dhabi, so fines, stranding, or return flights are not at issue.