In Firefox, for example, the search box remains open when moving to a new tab, but: it doesn't automatically perform the search. You have to perform an action to trigger the search: you have to press F3 again, or you have to click it again. This is not very different from with the current Opera: you have to press F3, and then Opera automatically performs the previous search.

Usually, Ctrl+click will open a link in a new [in the background] tab, and Shift+Ctrl+click will open a link in a new tab [in front of all other tabs].

I know of no extension in Opera that will cause a list of downloads to be displayed automatically after something was downloaded. Opera 12 had a download list which opened in a new tab (which was useful because you could get to it in the same way as you'd move from tab to tab). In the new Opera, you can open the Downloads list by pressing Ctrl+J (and you can customise that shortcut).

I also miss this vital feature. No satisfactory explanation has been given by Opera for this deficiency (other than the standard "for security reasons" excuse).

There is one extension that accomplishes half of what you're asking, namely:
https://addons.opera.com/en/extensions/details/open-with-application/?display=en

It saves the file to a temporary location and then opens it. In my case, I chose the Desktop as the temporary location, so every now and then I get a folder called somethi8ng like "temporary download folder" on my Desktop. If you click a file but then decide not to open it, the extension deletes the file from your hard drive again. This can be bothersome if you think you've downloaded a file, and then discover that the file isn't there where you've seen it just a moment ago, and you have to figure out where to download it from again (because unlike in Opera 12, the current Opera doesn't keep a history of all visited pages).

Hello
Since about a week ago, I've noticed that Opera no longer remembers the folder that it most recently saved a file in, but instead always saves files to the Desktop. Suppose I want to save a number of related images or web pages using Ctrl+S -- previously, I would create a folder on e.g. my Desktop e.g. "New Folder" and then when saving a file from Opera I needed to navigate to that folder only once. All subsequent files that I saved would be saved in that same folder. But now, Opera always chooses the Desktop as the save-as location, even if I've just saved a previous file to the "New Folder" folder just a second ago. How can I get the old behaviour back?
Thanks
Samuel

Since a few weeks ago, whenever Opera downloads a file, it creates a folder on my Desktop called "temporary_downloads" and then saves the file in there. How can I switch off that behaviour?

FWIW, this is what other programs on my computer look like (i.e. those that don't mess with the system colours):

Window with focus
Window without focus

Unfortunately Vivaldi doesn't respect system colouration either.

On Windows 7, version 45.0.2552.635 (PGO) of Opera pins itself to my taskbar, even if I unpin it. Every time I launch Opera, Opera pins itself to the taskbar again. When I then close Opera, it remains pinned to the taskbar. And every time I unpin it, and every time I run Opera again, it pins itself to the taskbar again.

Is this a bug or a feature?

Opera Reborn was announced. But the download links don't specify "Reborn". So... am I already using Opera Reborn (if I'm on the update cycle and I've recently installed the update which changed the window border colour)?

@joshl: To save a file in another folder, you have to navigate to that folder from the Save dialog. Opera uses the previous save location in the Save dialog, so before you can navigate to another folder, the "(1)" is already added by Opera.