Norwegian airlines restrictions--too onerous?

I'm good with their carryon limits so I can't comment on that. But I did buy a set of tickets for July with my Alaska Airlines Visa without even calling to give them a heads-up and the charge went through.

Too onerous? Nope.

I would never want to travel with more than 22 lbs so that weight limit common to many airlines doesn't restrict me. Next, the personal item rule is quite common and also is not an issue for me because my personal item is a tote bag fits under the seat. If your daypack fits under the seat, you're good to go.

I also read here that paying with a credit card can be a problem...so
what do you use if buying it in advance from the US?

What specifically is the problem? Next, from where did you read about this problem?

I am not an authority on this topic, but I can describe my experience flying on Norwegian from Iceland to Norway and from Norway to Los Angeles in July, 2017
First, it is my understanding that Norwegian uses Boeing 737-600 series planes for most European flights - that's the same plane used by Southwest. Their transatlantic flights are on 767-800 series Boeing Dreamliners - a much larger plane.
I flew to Iceland on the first leg of a three week holiday that included Sweden and Norway. I flew Icelandair from PDX to KEF.
When I flew from Iceland to Oslo on Norwegian Air, I was not permitted to carry on my backpack - I was forced to check it - but there was no additional cost for this service. That may be a polite policy or maybe I just caught a break. Ten days later, I flew from Bergen to Stockholm on SAS - another Boeing 737 - again, I was forced to check my backpack - again, no added cost. Maybe another lucky break.
I bought all those flights from home prior to the beginning of my holiday - no problem using my credit card.
I have flown Norwegian several times since then, but all have been transatlantic flights (on Dreamliners). Never a problem.
I'm flying Norwegian Oakland to Barcelona next month. They have become my favorite carrier.

I've flown Norwegian several times from the west coast to Europe, including segments within Europe. They will weigh your bags but if you don't think you can manage the 22 pound limit you can pay when you book your ticket to check the larger bag. It's probably about $25 each flight, but it's a lot more if you book it later or are required to check it at the airport.
If your daypack will fit under the seat you shouldn't have any problem with that.
I think that their website now states clearly that they weigh both pieces if you want to carry on, and that the total must be no more than 10 kilo,. although they didn't weigh my large purse on my flights.
It's a good airline and the rules are clear.
As to credit card issues, I think that on one flight Cap One had me type in a code they emailed me to validate the purchase. This happens occasionally on international buys so I keep my cell phone handy to check the code without having to leave (and screw up) the transaction in progress.

Their transatlantic flights are on 767-800 series Boeing Dreamliners -
a much larger plane.

Not so, from many of their east coast airports they use 737s across the pond.

Southwest flys 737-700 and 737-800 models currently plus a few 737-MAX planes. The -600 has been retired by Boeing as it was the smallest version of the 737 in recent years and was just too small for most airlines. Southwest never flew any 737-600s. They had nearly every other version of the pane and have added some of the MAX versions mainly for their international routes.

Norwegian flys 737-800 for many flights and have added 787-800 and -900 models in order to extend their flights further into the US past the east coast.