This year, I need a phone so that my husband and I can call each other, and for reservations, etc., in France- i.e., it has to work in France for 3 weeks, using French numbers-- and we also need to be able to call my son on his US phone, using his US phone number, which will have a "patch" so it will work while he's in France. I like the idea of getting a cheap phone here (Amazon) and installing a SIM card in Paris, making sure it works before leaving store. What phone do you recommend? What SIM card? What store?
You don't need a French SIM or French number to call French phones. Anyone with US phone service that offers international roaming (T-Mobile, Sprint, Verizon, AT&T) can use their phones in France with their US number but still call French phones. T-Mobile and Sprint both offer affordable roaming options (25 cents/minute calls to US and non-US numbers but free unlimited texting and free unlimited 2G data). Verizon and AT&T's plans are more limited and/or more expensive but will work. Many new phones also support "WiFi Calling" that will let you use those plans for free calls home to the US when the phone is on WiFi.
If you don't have one of those four carriers or just want to save money, you can buy a SIM ahead of time. The only French SIM I know that you can buy ahead of time is the Orange Holiday SIM (buy on Amazon). It is not necessarily a great deal and offers only two weeks of initial service (you can add credit - "top up" - to keep using it longer than two weeks). It would have the advantage of giving you a French phone number at which locals can reach you.
I use a Dutch Vodafone SIM that will work in France and is a better deal, but if you are not tech-savvy, perhaps I should not recommend it to you.
In Europe, a free app called WhatsApp is very popular for keeping in touch without actual phone numbers. You can install WhatsApp on your phone (do it before you leave for France) and even use it on WiFi without phone service to keep in touch with people.
As far as buying a phone: you can buy an unlocked smart phone before leaving the US. I don't know what the best deals are at the moment. It should be possible to find a decent Android phone under $100. The Moto E5 Play can sometimes be had for under $100 - it's a decent basic Android phone (make sure it is fully unlocked; some cheaper versions of it are not). I just bought a fully unlocked Moto X4 (Android) at B&H for $165 (still on sale for that price at the moment) and gave my old phone to my mother. The Moto X4 is probably much more than you need for just a travel phone, but it is a pretty snappy little phone for the money and will work on any US carrier and in France with any SIM card.
Also, can I use my "older" Apple iPhone like a tablet to access the Internet and email via Wi-Fi?
Yes. Any phone that has WiFi can be used in France just like in the US. You can even do WhatsApp on it and use it on WiFI.