I've done long term before, in fact, a couple of times and Hertz was great for this, IME.

They have weekly and monthly rates available. You'll get billed every month, with rewards being based on what you spend, just like normal. Rates will certainly be higher than, say, if you were to lease from a dealer. But you don't have the long term commitment, either, which was something I couldn't do at the time. If you have to return it early, you can, and they'll bill you incrementally. Can't remember what my rate was exactly, but came to somewhere between $350 and $400 per month, for a compact IIRC. This was about 2 years ago.

I used my personal car insurance. You'll want to check with your insurance carrier about what you're plan covers, but I think most that cover rentals do it as long as the rental is less than 6 months in duration.

Always better to rent at an off-airport location, airport taxes will kill you if you do it from there. The great thing is that they will also let you change cars, and you should be able to do that at any location, including at the airport. For example, I was given a Mazda 5 at one point from a small city location, since it's all they had. Took it to the airport a day later, and they exchanged it for a Corolla. Officially, I think they limit it to once a month, but in practice, not so sure.

FYI, I did check out a place that supposedly specializes in short term leases, but Hertz beat them price-wise by a lot.