The answer is no, maybe and yes… To answer your questions in order:
It isn’t vital to use a smoker, but it reduces bee stress and beekeeper stress.
Maybe you can put it off for a few weeks, but if you are worried about swarming, bee space or food stores, you probably need to go in anyway.
Certainly you can suit up and do it without smoke. If you are skilled and far from neighbors, it shouldn’t be a problem. If you are new, it might be very stressful for everyone.
Many people, myself included, find that after a couple of years experience they can work their bees without using a smoker at all.
Beginners are naturally anxious and a bit tense, which can result in jerky movements and a rushed inspection, squashed and agitated bees.
Taking things slowly and gently comes with practice, but you will get there in the end, and your bees will barely notice your presence (hopefully!).
A light spray with a water mister will calm the bees, and is far more pleasant for both bees and keeper. I use rainwater as out tap water is heavily chlorinated.