Yes only the Lighting cable or Qi wireless charger is compatible.
I was reading the product information but I didn’t found how much time I can listen to music or talk in the phone.
The information didn’t show anything about the battery.
AirPods with Wireless Charging Case: More than 24 hours listening time, up to 18 hours talk time.
AirPods (single charge): Up to 5 hours listening time, up to 3 hours talk time.
Yes. As per the product overview the charging case works with all generations of the AirPods.
Yes you can still charge it with a lightning cable
It appears the "Add Free Engraving" is only an option on the Full Sets of Airpods. Not if you're getting just the case.
You can see the option when you go to buy the Full sets of airpods.
"Add Free Engraving
Custom laser engraving available exclusively online."
I’m concerned that my AirPods battery performance will be affected if they are left sitting on my Qi charging mat overnight.
Yes as with any other apple product, over charging is not possible due to software management. Your airpods (or any other apple product) will charge to 100% and then trickle charge to make sure that they stay at 100%.
I got a pair of the first gen AirPods from my boyfriend and they are basically new.
No I do not believe there is any sort of discount for trading in your old case. As any other apple product, you can always recycle it for free. It's recommend just holding on to the case in case (no pun intended) you break or misplace the new one
I see replacement chargers available online but they specifically say “only for charging, not Bluetooth pairing”. Will the replacement allow me to pair with my iPhone?
Yes.
The button on the back is used to put your AirPods into pairing mode.
10w is plenty. 5w is also sufficient. I don't believe the AirPods support fast 7.5w wireless charging like the iPhones so it may not matter which charger you have
This case or any airpods case will charge any set of airpods. If your airpods are in the case, regardless of who's case it is, it will recognize them as your airpods. Thus switching to this case is as simple as putting your airpods in this new case, no repairing required.
I lost my Gen 1 case, and I am aware that the cases are interchangeable. Can I add engraving to the case when I buy online?