Hi Mugatu,
I've walked the Norte/Primitivo a few times in the range of that timeline, as that's when we have our school summer holidays and it's the easiest time to get a group of students on the road. We'll be back out there again this summer, starting from Irún on June 26.
Crowds are at their peak then and there can be a particular crunch in the first week through the Basque Country as there are a lot of people doing a short holiday. On the plus side, a number of summer-only albergues also open at that time, to help accommodate the masses. It's tight, but workable, especially if you avoid the major pinch-points.
It's hard to generalize from the past with great confidence on the weather front, but my experience has been that it's quite pleasant on the coast, often fairly cool in the morning, before heating up in the afternoon. It can get very warm in the afternoon, but the cooler evenings and mornings help to offset it. And besides, it's more fun to swim if it's hot.
Dave