Well, I have to write update here about my "Quest for multiple rows bookmark bar". After trying (see above), and also after trying current state of browsers like Maxthon, Otter Browser, Vivaldi I became sad and frustrated. Then I tried something I did not want to. I installed Firefox. Yes, sorry. Like I said - it was hard decision for me. But, after adding to Firefox two addons - Roomy Toolbar (it doesn't work well as Chrome extension under new Opera) and Speed Dial [FVD] + EverSync, I hate to admit, that I have finally Opera12-like experience, and even with good improvements.
I will check back time to time how Opera (new) and Vivaldi progress is going. For now I have what I wanted in place, where I did not want to look.