Glad we're making progress on the first point! But I think I know the error for the second point, it's a tricky one!
According to your Arcade expression, you have two values, right?
- "Inspected more than a year ago"
- "Recent inspection"
Even though you put these two values correctly into the arcade expression, the map still doesn't technically know that these two values exist... until inpsection records are submitted that meet each of the provided criteria (inspection within the last year OR inspection not in the last year).
So, if you built the arcade epxression and all of your inspections are within the last year, you can symbolize everything by "Recent inspection" because the map has data that supports this first part fo the expression. Until you add an inspection record that occured "more than a year ago", the map won't register this value in the symbology and it will ignore that second part fo the arcade expression. However, once you add an inspection record that meets the second criteria, you'll still have to go into the symbology and manually turn on the "Inspected more than a year ago" symbology. But it will work since a record meeting this criteria is available for the arcade expression to consume.
I ran through the symbology workflow and when I had no inspection records submitted, I couldn't use either of your two values. I got a generic "Other" value. I added an inspection record with today's date and was able to add the "Recent inspection" value. Then, I added an inspection record and falsified the date to be three years ago; this allowed me to add the "Inspected more than a year ago" symbology color.
Here's my suggestion: when using an arcade expression like this to drive symbology, create "fake" inspections records that will trick the map to thinking that all of your different criteria are met. This way you can apply your: "Inspected more than a year ago" and "Recent inspection" symbologies.
Finally, it makes sense that the symbology from the Map Viewer Classic pulled into the New Map Viewer. Just like the utility symbology, the New Map Viewer will pull on that Map Viewer Classic's symbology.