Anytime.
I am a recreational rock climber. I've climbed outdoors a few times, and up to 5.10a indoors (though only once). I'm not nearly experienced enough to undertake a new climbing route entirely on my own, but experienced enough to climb outdoors comfortably. I've done Ice Climbing as well, but I figure that won't be possible in Pakistan after March.
You wont find much here then. The organisations that would take you rock-climbing won't take you to anywhere of note. I would recommend scrapping this. Btw, you can still ice climb after March here, GB is the most glaciated region outside the poles after all. But you would face the same problems as with rock climbing, tons of opportunities not enough facilitators.
I've never done paragliding. I'm looking for a tandem beginner flight somewhere quite scenic, such as Skardu.
Sadly, none available in Skardu.
The problem with adventure sports in Pakistan (except mountaineering of course) is that to go beyond the elementary level (which are quite meh) you need to do it yourself. Won't find much help here.
At this point, I'm looking to do camping, day hikes, swimming, paragliding, climbing, fishing, and whatever comes my way. I don't really have the time to do any long expeditions this time around.
Here's the map I have of the places I'd like to visit. You're right, I've never been North of Islamabad before.
https://drive.google.com/open?id=1GJpueQdh41elsCgL1vGxBDp46Y0&usp=sharing
Ok so let's begin. The thing about Pakistan, especially the north, is that there's too much to do and even more to see. Although your list includes places that just can't be covered in a single trip, we can try packing as many of the places and activities mentioned by you as we can into a single 1 week long trip. I usually recommend travelling nonstop to Skardu/Gilgit but we'll take it easy, it's only your first time. I would recommend picking either Skardu or Hunza as your ultimate destination. For now we will go with Hunza as the example (If you'd rather make it Skardu, let me know). Would you be staying in Lahore? If so then you can check out the mosque, the Minar-e-Pakistan and the fort in a single day and then maybe dine for dinner at Cuckoo's Den (It's not as good as it used to be but you can't beat the view). You'd have crossed two places off before even beginning.
Starting from Lahore I'd recommend leaving the afternoon before around 2pm(You pass through the Kaghan Valley during day time this way, you'll thank me later). You'd take the motorways till Hasanabdal (burhan interchange). You can stop in Islamabad (4hrs 30min from Lahore) for an early dinner but you should be on your way to Hasanabdal before 8pm. After you leave Hasanabdal you'd pass through Khanpur before you reach Abottabad. This is where you can go paragliding, but forget it (again you'll thank me later). The scenery will have started to shift before you reach Abottabad, but it's only the beginning. In Mansehra you'll have two route choices; towards Badgaram (the classic KKH route through the Kohistan region) or towards Balakot (the newer route through the length of the Kaghan Valley). It all depends on the season you go. I'm hoping that it's after April, you get to see both the routes this way. If so, then you should get off the KKH at Mansehra and take the road to Balakot. If you would not have stopped too many times up till now, you should reach Balakot at around midnight. You can choose to stay the night here, there's a chaplu Kabab dhaba on your right just before you cross the bridge, I recommend.
Welcome to the North. Next morning you should be up and ready to leave before light. I'd recommend grabbing some quick snacks and delaying the breakfast. From here onward the drive itself will be a treat. From Balakot it would take you around 3 hours to reach Naran, if you leave at 6am then you can have your breakfast in Naran at around 9am. The town isn't what it used to be but you'll get to cross it off your list. You could hire a jeep from here and to take you up to Saif-ul-maluk (2-3 hrs round trip) but I'd recommend ditching that and saving the time for far greater places further ahead. You should be on your way again before 10am. The route will then take you through Batakundi, Jalkhad, right around Lake Lulusar (pretty spot) and then to Babusar Pass. Babusar is extremely beautiful and it would take you around 2 hrs to get there from Naran. Crossing the pass you will exit KPK and enter GB's Chillas district.
You will notice a very sudden change in scenery. The vegetation will vanish and the mountains will start growing even taller. You'll link up with the KKH here and reach Raikot Bridge in another 2hrs (2pm). If you were to visit Fairy Meadows, this is where you'd hire a jeep for it from. But we will keep on driving. A little further you'll reach the Nangaparbat view point. It is truly spectacular and would change your perspective on mountains altogether. Moving further, in about an hour from Raikot, you will reach Jaghlot. You can stop here for lunch. Just after Jaghlot you will come to the junction point of the three highest/greatest mountain ranges in the world. You'd be driving on the side of the Hindukush (on your left), on your right would the Himalayas across the Indus and straight ahead of you would be the Karakoram. Just a little further from here is Alam Bridge, if we were going to Skardu we would have crossed it eastwards over the Indus and entered Baltistan. Leaving Jaghlot at around 3pm you should reach Gilgit before 5pm. You can roam around the city a little, shop a little, eat a little, it's not much. You can also spend the night here if you are excessively tired but I'd recommend sucking it up for just another 3hrs and reach Karimabad.
Congratulations, you are now in the Hunza Valley. There are a bunch of different hotels you can stay at here. Spend the night here and wake up at your leisure the next morning. I'd recommend a hearty breakfast at Cafe De Hunza. Keep this whole day to enjoy Karimabad. Any direction you'd look, you'd see a +7000 meter peak. You can check out the Altit and the Baltit forts. Go up to the Eagle's Nest and take in the 'knock your senses out' birds eye view of the Hunza Valley. Shop for handmade exotic items in the market, check out the many different fruit orchards or even drive the very short distance to Aliabad, there's tons to do. TBH, if you just sat your entire day on a chair in your hotel's yard here, you would not regret it.
Now is when all your activity based options open up. There are too many day treks available here for me to count. Your hotel manager will be more than glad to recommend a trek according to your mood at the time and arrange guides and porters for you. You can ask the locals for prime fishing spots too and they'll happily show you around. I would recommend going on a day trek the 2nd day and then leaving for Khunjerab Pass on the 3rd. You can leave early the third day, have lunch at Sost, reach Khunjerab by around mid-day, return to Passu for the night and then drive back to Karimabad before mid-day the next day. Spend the rest of the day relaxing in Karimabad because the next two days are going to be the long drive back completing our week long trip.
With this tentative itinerary you would have crossed off Badshahi Mosque, Minar-e-Pakistan, Lahore, Karimabad, Gilgit, Naran, Kaghan, Hunza, Mountain Ranges Junction, Khunjerab Pass, Khunjerab National Park and a ton of other places which you don't know, yet, that you should have on your list.
Let me know if you need any further help, but to recommend you anything further I'd need more details of your trip. I hope I've at least been of some help thus far.