Public transport from TLV is cheap and it runs often - but it isn't always the right fit for every trip. Here's an honest comparison.
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.