How can I add time with a prepaid card?

It depends on the gas station. Some will not let you use a credit card to buy a prepaid card.

Yes, as prepaid credit cards act like regular credit cards.

You are basically transferring cash from a "permanent" card to a temporary, disposable one.

You can use the internet to check by opening any app that runs on the internet, like Whatsapp, Spotify, YouTube, the browser for any online search, etc. If it doesn't...

By inserting it into your phone. It should be automatically activated upon usage.

With the prepaid card, you should be able to watch Netflix on any device without a credit card for as long as the card can pay for it, yes.

It depends on where you are, but most European countries carry ones in grocery stores that you can just buy.

Yes. Most SIM cards bought in Europe offer Europe-wide connection.

You can buy the so-called "unlocked" phone that has the GSM network built-in and then when you get to Europe, buy the SIM card at the airport.

Yes, they are the same. Rewards cards are just because you spent money in a store, while gift cards are something that people give you.