2-week Barcelona and Basque country trip (August). I'd like some critique of this itinerary.

Pamplona can be seen in a day, so unless you plan on visiting nearby places like Olite, Estrella, Puente la Reina, Alto de El Perdón, Foz de Lumbier, etc., then one night is sufficient.

Burgos is always busy with pilgrims as the Camino de Santiago goes right through its center. There are nearby places worth visiting, most notably, the Cartuja de Miraflores. If you are a Clint Eastwood fan, Sad hill cemetery from The Good, The bad, and The Ugly is by Santo Domingo de Silos. By itself, one only needs a 1-2 days in Burgos.

If you will admit Jaca then you could...

Take the train to Zaragoza. You say summer so it may be hot, but it is worth the visit. Then rent a car, go to Huesca and Jaca. Visit the Loarre castle and San Juan de la Peña monastery.

After Jaca you can go in to France. Pau is nice. Next on to San Juan Pie de puerto. Dont cross at Valdecarlos rather go to Baigorri and enter Spain at the Izpegui pass. Spend the night in Elizondo. There is great hiking in the Baztan

Later head over to Bilbao. Leave Bilbao and head to Logroño. After your stay there you can leave the car and train to Barcelona. Or drive to Zaragoza and take the train.

Skip Pamplona. It is worth a day but not three. Burgos is very nice, but it is a bit out of the way for your trip.

When in summer? You may hit some fiestas. And how many?

You could consider Montserrat by train one of your Barcelona days- leaving early from the city and hiking up as soon as possible. Then return to Barcelona by early afternoon. It was July and yes, very hot, when I was there! The hike was great.
The Basque region, both countries, was one of my favorite trips ever. I stayed in St Jean Pied de Port one night and although dismissed by RS in his guidebook, I found it charming. I still have my seashell, and just hiked the Camino from that starting point about an hour and then back.
I did a day trip to Sare from SJ de L and the hike back down to the bus stop. Warning to self (and maybe others!)- it was a beautiful, clear day on the way up via the cog train. As I started down, what should have been easy rapidly changed as a fog so thick I literally couldn't see 2 feet ahead of me rolled in from nowhere. Hearing the horse bells was both haunting in a good way (later) and a bad way (during)!
It was definitely cooler temps in that area, even at the coast.
Have a fantastic trip!

The side trips from Pamplona mentioned above look nice. The first night in Pamplona involve traveling over the Pyrenees into France then arriving in Pamplona; that leaves 2 more days and can be split into touring Pamplona itself and side trips.

Likewise the first night in Burgos can involve driving from Bilbao to Santillana del Mar and Altamira cave then arriving in Burgos in the evening, leaving 2 days for Burgos and possible side trips. Biggest downside of Burgos is the return drive to BCN is kinda longish. My flight doesn't depart until 7 pm, so it's doable, though I'm considering splitting the 3 nights into 2 in Burgos, and 1 final night in Zaragoza or Tarragona for short approach to BCN.

Don´t expect sunny and beach weather every day on the northern coast, it´s rainy and humid and with very unreliable weather. But absolutely beautiful landscapes. Pamplona deserves half a day, maybe one day, with side trips to where dilindstrom has suggested. Bilbao is worth the stay, of course, as it´s central location to many other places in the area (Gernika, Bermeo, Mundaka...). But again, come prepared for any kind of weather!

We will have rain gears (due to the planned hike). I'm not concerned about rain actually, and much more re: heatwave. I think that will be much worse. I can shorten Pamplona to 2 nights, so 1 night for driving across the Pyrenees and touring French Basque towns, then another day for Pamplona itself. I can keep Bilbao and Burgos at 3 nights. That leaves an extra night to break up the drive back from Burgos. I can stay in Zaragoza or Tarragona.

Heat waves are very rare on the northern coastal areas of Spain, so it´d be something exceptional during your visit of Bilbao and surrounding places. More heat in Burgos and Pamplona, no doubt, but nothing like the south of Spain. Temperatures in the mountainous, lush green Basque Country area are much cooler and you may need a light jacket at night. And raingear, just in case.

I expect Burgos and Pamplona to be hotter for sure, but their night time temperatures drop down quite a bit, so they are probably okay unless I hit a heatwave. AC is good to have just in case but not essential. If I stop by Zaragoza/Tarragona on the way back to BCN, I'd definitely book a place with AC.