Whether an iPad (Pro or otherwise) can replace your laptop is dependant on 3 things.
- What do you need\want to do?
- How flexible are you about your workflow\how you do things? (If you want to do things in the exact same way as you do on your laptop, then the iPad will most likely fail.)
- Do the right iPad apps exist to do what you need\want to do? This included things like multitasking support. (Which I find works well on some apps, and quite poorly on others.)
For me, there are a couple of Windows only apps that are critical to my day job, so I will continue to need my work supplied Windows laptop for the foreseeable future. That's not the iPads fault, the apps could easily run on an iPad with an optimised touch GUI (In fact they are horribly designed, and could be greatly improved with an iPad style interface), but they have chosen to only support Windows. And outside of work, there are times and applications, when I just prefer to sit at a desk with a much bigger screen. (It makes me laugh when people say the 12.9" iPad is huge!) But my iPad is by far my most used computing device outside of work, and as I discover ways to adjust my workflow, and find the right apps, I expect I will use it even more. However I expect to keep my desktop for the foreseeable future.
If my memory is correct, the iPad Air 1 doesn't support split view, so upgrading to a pro would allow you to use Safari or Notes and Skype side by side. The Air 1 only has 1GB RAM, while the 9.7 Pro has 2GB, and the 12.9 has 4GB so you can have more things open without running into memory constraints.
Of course there is nothing to stop you from using a "proper keyboard" with your current iPad, just buy the bluetooth keyboard you like and connect it. An iPad Pro adds the option on keyboards with a Smartconnector, which is nice. I doubt that Apple will ever add mouse support to iPad, (Sometimes I wish they would) so if you can't adjust to not having a mouse iPad isn't for you.
So to answer your original question. Yes iPad Pro has addressed some of your issues. Split view multitasking is definitely improved over the Air 1. But some of your other issues are more about the software, which no new iPad will fix, Pages is still just Pages on an iPad Pro.