I'm sure if you ask the airlines, they'll say no, that you can't check it if there isn't a child with you. BUT I would just have your hubby take it with him and if he's questioned about it, just say that his wife's already on board or in the airport. I'm sure no one will question it more than that because most won't really care. I would suggest that he checks in online with the number of bags he's taking and not include the car seat. I would recommend taking the car seat with him up to the gate and then gate checking it there (down the little corridor that attaches to the plane - there's usually a collection of car seats and strollers there waiting to be loaded by the baggage handlers). They also load those last and take them off first, so I find that the likelihood of them getting bounced around excessively or spilled on is less than if checked when you arrive to the airport (this is just from our personal experience).
As to the others asking questions, it seems that as long as your husband is arriving prior to you, then the car seat would already be in the car when he comes to pick you and the kids up from the airport, so I don't see how that would be an issue. And to those asking about needing the car seat to get to the airport, I would imagine that you (like most families with two parents) would have at least two sets of car seats and he would be taking the alternate set, so you'd still have a set to get to the airport.
I am HIGHLY opposed to using car seats that you don't know the history of, rental ones included. I don't trust that the company or individual letting you use it would tell you honestly if it had been in an accident - which is why I would never consider purchasing a used car seat from ANYONE. Also, many people who could be renting these things out might not know that if it's been in an accident that it should be replaced and they might not know anything about expiration dates (so then you could be stuck at the airport if it shows up with an expired seat) - and how would you know if the rental place purchased their seats used, furthering the idea that you have no idea the history of that seat. Strollers for rent, I don't see problems there but definitely with car seats.