I have a 20mo and a 23mo right now, so I've just been through this stage twice in rapid succession.
A lot is going on at this age, but one thing I always have to remind myself is they are usually teething. And teething does things to people. Like make them insane. Think of it as a toddler migraine. Then, they are growing and learning rapidly, which makes them tired, and they are noticing things about their body such as elimination, which they don't like being unable to control. I also noticed that my kids developed strange fears at about 18 months - fears of things that they thought nothing of before.
I am very strict, but I find that sometimes, the little one is just hurting and doesn't even know what her problem is. At times like that, is it really that important to make them walk down the stairs? When they are hurting or afraid, arbitrariness will only make things worse. A little kindness followed by a reminder that that behavior isn't necessary, etc., was much more likely to calm my kids down and get them back on track. I usually didn't get repeat performances because it really wasn't a power play at all.
We worry about the child "winning" but are we really in competition with our kids? I am not sure why you don't want to carry your child when asked - maybe you are carrying a load of laundry or another baby - but if it's not a burden, I'd just do it if it seemed that important. You wouldn't refuse to give your child a requested hug, would you, even if it wasn't in your original plans? I promise that even if you carry your child a few times, he'll go back to walking the stairs eventually if you don't make it a battle.
If the child simply wants something she's not allowed to have, I'll ask the child to confirm that's what she wants ("you want this pen?"
, so the communication frustration isn't there, but then say, e.g., "you can't play with the pen today, maybe another day. I'm going to put it away." This has worked for me a lot better than an arbitrary "no."
When things got really out of hand - e.g., screaming over some small thing for more than a half minute - I put the offender to bed for a time out. I didn't go by the recommended "one minute per year of age" but left her in there until she started acting human again, which took about 15-20 minutes a couple of times. This was quite effective, but I wouldn't want to do it for every small offense.