Comparing taxi, train and bus from Ben Gurion Airport to Tel Aviv and Jerusalem - price, comfort, luggage space and Shabbat schedule, so you can pick what's right for your trip.
Often yes, door-to-door - the train itself is quick, but you still need to get from the station to your final address. A taxi skips that extra leg entirely.
You can, but you'll be carrying it yourself through the station and onto the platform. For a family with multiple suitcases, a private taxi is usually much easier.