You don't need to. But it will allow you to press more weight with less risk of injury. It only feels unnatural because you're used to just lying flat on the bench with a relaxed back - doing that now feels "unnatural" to me.

Try this: walk over to a nearby wall and "try" to push it over. That is, put yourself into the position that feels like you have the best brace and power to transfer force into the wall, and actually do it. Notice where your hands, elbows and shoulder blades end up. That's basically the same position as a good powerlifting bench.

Scapula should be firmly on the bench. They're responsible for transferring the load to the bench.

If they're out of position, you'll get hurt.

With decent form and a reasonable balance of pushing and pulling movements (don't neglect your back!) you'll have healthy shoulders.

It took me over a year of benching before I realized that my shoulders shouldn't feel anything from benching correctly. There are a ton of good youtube videos on scapular retraction, if you learn to do that and make it part of your routine you'll never have shoulder pain again. I'm not sure if it makes benching more or less efficient, but I'm able to recover significantly quicker and so I've made better progress since learning to retract.

Yes. But like all things, it takes time and practice doing something different before it feels natural. Doing it this way is better for your shoulders.

Good idea to do this during squats as well, but for a different reason: engaging the lats (the "pulling my shoulders down" part) helps to make your back more rigid.