Responsibility in a multi legged flight itinerary involving multiple airlines

It's hard to tell a generic answer here.

1. This rules governing this are typically spelled out in the terms and conditions (T&C) of the airlines. Without knowing what the specific airlines are, we can't help much. I suggest reading them yourself (you did agree to them when you booked).
2. It's not unusual to get stuck in "finger pointing". I.e. the T&C of airline A will state that B is responsible and vice versa. Some airlines are better than others.
3. In most cases the operating carrier is on the hook: it's their fault and in some cases the marketing carrier doesn't even has a presence at the location where the problem happens. I recommend being insistent: "it's your flight, your cancellation and hence its your responsibility". Rinse and repeat. Try a different agent if the first one isn't helping out.
4. I wouldn't get my hopes up for engaging a "higher authority". If the issue happened in the US (and the authority is not of the religious kind), you can try https://www.transportation.gov/airconsumer/file-consumer-complaint

Even with the additional information you've provided it's still difficult to give a complete answer, but let's try...

I'm going to presume this issue occurred within the past few days. The US is currently experiencing a 'once in a generation' weather event. Well over 10,000 flights have been cancelled across the country over the past few days, and airlines will likely be struggling to catch up over the next week or so. Unlike most storms that affect only a part of the US, this one has affected every single US mainland state except California, has resulted in countless state-of-emergencies being declared across the country, and even resulted in some airports being closed for days on end.

This is almost certainly the cause of your cancellation/delays. The specific reason might have been unavailability of crew as you've stated, but that will be a flow-on effect of the weather.

In a situation like this, there is no requirement under any US legislation to provide any form of accommodation, meals, compensation, or pretty much anything more than a commitment to eventually get you to your destination as soon as if feasible. The airline themselves will also not offer any of these things where the delays is being caused (directly or indirectly) by such a major weather event. (There are a few exceptions here, such as provisions by some airlines where you are flying international business class - but you haven't mentioned that you were, so they likely don't apply).

The government organization that you would need to complain to would be the US Department of Transportation, but given the weather event currently occurring, the odds of them doing anything with your complaint is zero - especially as it doesn't appear that the airlines have actually done anything wrong.

As far as which carrier is responsible for getting you to your final destination, this would primarily fall to airline B, presuming that you have already completed your trip with airline A, AND presuming that airline A delivered you to your connection point on time. Rebooking for "Day of flight" cancellations are the responsibility of the airline that canceled the flight. Airline A (being the airline that issued the ticket) will likely also be able to rebook you in this specific case. Without more details it's difficult to know which option would be best, but personally I'd probably be trying both options.

The correct course of action to take at this point is to contact your travel insurance provider, as this would almost certainly be covered by travel insurance. This, of course, presumes that you have travel insurance...