As far as the rental car goes, it really depends on how far away from the parks you are. If it's a Downtown Disney hotel, you can get by without a car. But for me, if it's even a mile or two away, I'd rather have a car even for just if I wanted to run to a store for something (or go out to dinner cheaper!). So, for the most part, you may want a car. I'd look into the car for the whole time, if it adds up to a week it may be more reasonable to keep the car the whole time. Even if you don't use it while staying on property, you won't pay for parking it at your Disney resort. Plus, you can do a grocery/snack stop if you want.
I've also rented cars from
Alamo/National for parts of stays, as I like that you can pick up or drop off at the Dolphin. The rates do seem to have gone up a lot over the last year or two, so play with codes and rates (I like mousesavers for help with discounts).
In taxis, we have never used car seats when our kids were toddlers. Usually we're not going far. For the airport towncar service, they can provide carseats and boosters, but I don't know about taxis. If you feel more comfortable, you can bring one from home for a rental car. I don't think taxis require them.
Is your offsite stay at the end of the trip? If you don't mind telling, where it is? I know when we stayed at Marriott's Grande Vista, even though we had a car, we split up twice, and I took a cab with the kids to the MK, and it was about $50! Yes that's right. I sucked it up, since it was vacation and we had saved so much on the condo and tickets (military deal) but I had thought it would be more like $30 or $35. It's further than it seems. Even Marriott's World Center is probably at least $15 or more each way, so it can add up.
I hope that I've helped, not made it more confusing. It's early and I'm still drinking coffee!!!!