Sparring is an essential part of training in martial arts. It's done differently in different styles of martial arts and while the techniques can differ substantially, its basic purpose and goals are very similar. Its main purpose is to prepare you for a real fight. You should bear in mind when training that the goal of sparring is not to win, but to improve.
I find it particularly useful to spar against someone who is better than myself. Of course both partners can't be better than each other in every area so there is always some degree of compromise. Consistently sparring someone who is worse than you will hold you back as you won't be pushed to your limits. Sparring people from other martial arts schools and other styles is useful because you can compare your skills against someone who was trained independently.
I will now list some of the benefits of sparring which will hopefully highlight why sparring is so important. There are of course more, but I single out these as the most significant.
Certainly not. Punch bags are good for learning, practising and improving your techniques, but the problem is that punch bags don't move and don't fight back. Sparring is all about blocking and counter striking as that is what a real fight would consist of.
How hard you should spar depends on what you're trying to achieve. Most martial arts schools spar a bit harder than light sparring. Hitting your opponent hard really doesn't achieve that much apart from unnecessary pain. On the other hand a real life fight would be hard. Be aware of the dangers of sparring.
It depends on whether the sparring is light or full contact. You can get away with no protective equipment, but I would recommend sparring gloves at the very least. At the other end of the spectrum you have head guards, gum shields, body padding, shin guards, foot guards, etc.
Yes. A big problem. By not sparring you're not preparing yourself for a real fight. I would watch out for martial arts schools that never spar as quite frankly they're a waste of money. Any experienced martial artist would recognise that sparring is essential. I would be suspicious if your instructor is not letting you do it.
They may not allow sparring because they can get away with doing easier activities. They could even be afraid of the consequences of injury. Who knows. But do yourself a favour. Next time you're there, ask the instructor why they never spar. If you're bombarded with pathetic excuses then perhaps you should look for a better martial arts school. There is no excuse for a lack of regular sparring.