You can't use cntrl + number the way Opera is set up unless you can find an extension in Opera or the Chrome store that does. I took a quick look and didn't see it. You know, as long as you have downloaded the Opera extension, "Download Chrome Extension," you can also use Chrome Extensions in Opera.
On your question about tabs, the help section on Opera gives the keyboard shortcuts.
http://help.opera.com/opera/Windows/1387/en/fasterBrowsing.html#keyboard
So you would have to get used to control tab to go forward, and control shift tab to go back. But in the alternative, you go to Settings, and Enable Advanced Keyboard Shortcuts, you will see that you can cycle left with just a "1" and right with a "2"
As for Extensions, I looked at Tab Manager in the Opera store, it does not use the control 1, 2, etc. However, if you often browse with multiple windows of Opera open, it lists all of the tabs in both windows, making it easy to go back and forth between the different windows. There may be an extension that uses control 1, 2, etc. You would have to look for it. If not now, it may come in the future. I'm not going to use that extension myself, as I don't want to overload my browser with things I don't really need.
On the search bar, did you try a google search through the address bar? as well as through the google search bar on the Screen Dial page? Did they both go through google.rs. I can't check that on my browser, because I eliminated the google search bar (through some browser supported customization), and for the address bar, I am using the Disconnect Search Extension, that routes all Google Searches through Disconnect, to protect an individual's privacy in his searches.
I hope some of this might be of help. I'm glad that you are in general happy with Opera!!! I enjoy the browser very much.