There is much to touch on from the comments so far, but I'll try my best to keep it on point. (Note that I didn't say "short" <wink>.)

Bona fides: FAA Licenced Aircraft Dispatcher; 11 years industry experience. (Left the industry in 2000)
Staff title: Chief Dispatcher

Preface:
I'll not attempt to address the many meteorological or airframe engineering aspects of this mission other than to note that Delta staffs its own meteorology department (or did last I was privy to their operations). The capabilities that lend to the carrier are net positive, as should be obvious.

On to some specific questions raised ...

(0)
>>>[qume] The captain makes the same call on every flight. The plane and passengers are her responsibility regardless of the situation.
Edit - also the call is made continuously. They can back out any time.

Response:
Only half true: the pilot-in-command (PIC) along with the aircraft dispatcher share responsibility for the "initiation, operation and termination of the flight." (Yes, I think the regs use 'termination'; that always made me wince.)

So, it's a quorum of two: if either one chooses to terminate (or 'not initiate') a given flight, it cannot be operated. That doesn't mean that a dispatcher who disagrees won't spend some effort presenting evidence for his position (e.g., ten-minute phone calls), but at the end of the day of those two don't agree, the flight cannot operate.

(1)
>>> [phkahler]
They may or may not have volunteered for the flight, but they do get the final decision to go in or turn back once the airline OKs it.

Response:
This statement begs the question: who is "the airline?" The relevant US regulations (CFR 14 Part 121) refer throughout to this entity as the 'certificate holder' -- because an air carrier holds an operator's license -- a business license, of sorts -- granted by 'the administrator,' the regulatory term for the FAA. I'd gamble that since the FAA descended from the CAA that the policy authors thought it wise to anonymise the parties wherever possible in the event of future language changes. Smart move.

So for any given flight, "the airline" would be the dispatcher, who by proxy, exercises the right of 'the certificate holder' to "operate a particular flight over a specific route under specific conditions" (going from memory, mind you).

This authorization is formally granted by way of a legal document prepared by 'the certificate holder' (read: dispatcher) known as the 'dispatch release,' which includes a minimum set of specific information (flight plan, equipment type and number, flight crew, fuel min/max/burn, alternate airports, and so forth) but typically have an abundance of supplementary information to better brief the flight crew of the expected conditions and details of alternatives that are likely to be available if the proposed plan cannot be followed.

Bonus fact: If you ever were waiting after boarding and the crew came over the PA to say they are "waiting on paperwork," most of the time it is a bag/weight count, but some times it's the dispatch release. If you know your flight is going across, or into, some crappy weather, the chances of the latter are greater than average.

(2)
>>>[passivepinetree]
Does anybody who works in the industry have any idea of what the risk management is like for these types of events?

Response:
The airline operations hundreds of flights a day; all have some risk, and all decisions must be made in real-time. In cases of long-running events such as a hurricane, there is likely to be some general tone taken by the carrier at the highest operational levels (chief pilots, chief dispatchers, VP of Operations, etc.). For my carrier, these strategic positions would be discussed in the morning meeting, which was a recurring conference call between all those parties and department supervisors -- kind of like a stand-up, except we were all sitting around a conference table.

Aside from that, as the day wears on the specific handling of a given route, weather event, etc., is handled in real-time by the assigned dispatcher and support team (meteorologists, mechanics, etc.)

Of note: some larger carriers maintain a 'Trouble Desk' staffed by dispatchers who are assigned a lighter workload than the regular line folks. This is a great system (one that my carrier didn't have) because, let me tell you, just one fubar flight can monopolize all your capability and time for quite a while. If/when one of those flights pops off the queue it can wreck your throughput for the remainder of the shift. For my money, the trouble desk is an excellent mitigation tactic capable of keeping the workload in the dispatch office sharded appropriately.

(3)
>>>[joemi]
Does anyone know more about this type of pre-hurricane flight? Is that a common thing airlines do, to try to squeeze one last flight in? Are there rushes before other types of bad weather?

Response:
I only had a handful of duty experiences with hurricane landings that were in our region of operation, but I do have a wealth of experience with other severe weather systems here in the US -- aka, tornado season.

There were many occasions when a strong cold front would be bearing down on cities we served, with solid lines of thunderstorms sweeping through the region. The kinds of weather that serve up severe or extreme turbulence, large hail, and tornadic activity. And there aren't any "holes" to "slip through," either.

Frequently, it would come down to trying to get one more flight in and out of a city. (The pressures of 'completion factor' at an airline are a whole discussion in itself.) In these cases, I'd be measuring the relative velocity of the line versus the distance to the airfield, estimating the time of impact, so to speak, and cross-referencing that against my computed time en route and considering the turn-time at the station, etc., etc.

Assuming you judge that it can be operated safely, and you can provide a suitable alternate plan (and "turn back to base" is certainly a common choice) you have to get on the phone (or radio) and brief the PIC on some or all of the details (all of the details are in the dispatch, but people like to hear a human voice when facing stressful situations; think 911 operators), and if they concur (or accede, in some cases) then, from (1), by the necessary joint agreement, the flight is initiated.

And as you might expect, sometimes the flight got in and out, and sometimes it diverted or came back to base. In either case, occasionally the crew might call or radio back in with reports on the conditions (or vociferous complaints about the ride quality -- hey, it happens).

So then my unqualified answer to this question from joemi is "Yes."

--

Finally I'd like to call out the commenters who made mention of the ground (and other station) crew and their exposure to risk in these "irregular operations"; I'd say they exhibited a measure of aplomb no less than the flight crew, and those employees are too often overlooked as essential parts of the carrier's operations; both day-to-day and in extreme cases such as a hurricane landfall.

HTH,
/Acey