You do not have to have an infant carrier if you don't want one, though many parents prefer the convenience of them for the first several months. The hospital probably will require that you have the seat, so you may have to bring it in or have them see it in your car.
It is NOT okay to get a used car seat. Car seats are one time use and if they are in an accident must be replaced. Never buy/accept a used seat from anyone you would not trust your child's life with. You do not know how that seat was treated or what it's been through. A used seat is also more likely to be missing the instruction manual and/or important labels to help you determine if the seat has been recalled or is expired.
The expiration may be listed a few different ways. Some have a sticker that says 'do not use this seat after xxxxx" Others have a kind of clock looking thing molded into the plastic. If you cannot locate either of these, check the model sticker. It will have the model number, serial number and date of manufacture. Then check the instruction manual for how long the seat lasts. 6 years is the most common expiration, so if you find nothing else, go by that.
You should not have to prove that the seat is new, but it sounds like yours is, and I would not get a used seat if you get an infant carrier, so that would be new, too. They cannot require this because many people will use the same infant carrier for a second child. They may check that it has not expired, though.
You said you got the seat you have planning for it to be the only seat you'll need. You have a 3 in 1 seat (my 1st guess is the Alpha Omega Elite or Eddie Bauer, though I couldn't find either for only $100 so it may be a different one) and though these advertise to be the only seat you'll ever need, it's just not true. First off, they expire after 6 years and you already have it before your baby is born, so it's going to expire before baby turns 6 and that is too young to be in the vehicle without a booster. Second, they tend to not fit kids well as a booster, so you would need a dedicated booster anyway.
If you're set on the 3 in 1 seat you have, it does make a fine harnessed seat, both rear facing and forward, so you can use it for several years before you'll need to look into getting a dedicated booster.
I, personally, prefer convertibles because they harness to higher weight limits than the 3 in 1's. Most 3 in 1's go to 35 lbs rear facing and then to only 40 lbs forward facing (the 100 lbs listed is for booster mode). Most convertibles rear face to 35 or 40 lbs and forward face to 65 lbs (some as low as 40 lbs like the 3 in 1, some up to 80 lbs)
A higher weight limit is not always necessary, it completely depends on the child. Some will outgrow their seat by height long before weight, so the 3 in 1 would last them just as long. But if you think you may have a chunky little one, you might want a seat that has a higher weight limit.
I hope I answered all your questions. Car seats are my passion and my obsession! Feel free to ask me anything else!
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