Deactivating hardware acceleration finally DID THE TRICK! Videos are working -I've tried to play same videos with acceleration turned off and on, and every single video that was displayed as blury "something" before is now playing perfectly, also no issues when enlarged.
It seems that blank tabs are gone too. I can't 100% confirm, because as I said before - this was very random, but I've tried to click on a lot (like really a lot) links and again - there was any single blank tab amongst them. Also clicking on "Community" on main Vivaldi page would gave me blank tab in 9 cases out of 10, now it's always opened and displayed.
That thing with GPU is indeed weird. As far as I remember, I had only one major issue with GPU since I have this notebook (2013) and it was with game The Witcher 1. There was some kind of launcher testing HW before game was actually launched and it kept saying that I have only Intel HD and it's not enough for a game so I had to modify .exe with some parameter that allowed to skip that test and launch the game, however it forced me lowest possible texture resolution, even if the game clearly started using Radeon once launched.
If whole thing with blank tabs and videos is caused by drivers I guess I can hardly except that AMD is going to fix it, because last driver on Radeon HD7650M is from 2015 (both on AMD and HP websites).
Anyway, thank you a lot for your help and keep doing great job. Looking forward to future releases.