I switched yesterday from NPAPI to the PPAPI flash plugin without any problems. Working fine so far.
You might consider to uninstall the NPAPI flash if PPAPI is working fine for you. You can see the detected plugins here (click on show details at the top right corner):
opera://plugins
Nevertheless Opera wouldn't load both plugins, but prefers PPAPI as far as I know.

I tried this too, but it has brought no joy.

It is so strange. I have a flash game open in one tab right now and when I go to this tab to type this message, it is as though my game is paused.

  Are you using ppapi flash or npapi flash?
    

NPAPI.  What is the difference, if you don't mind sharing some knowledge with a less fortunate soul 

Try the ppapi one, it uses to be better.
Ppapi (or pepper) is a new plugin architecture, more modern.

Choose the third link under the 'Windows' heading about halfway down the page, titled 'Download Flash Player for Opera and Chromium based applications – PPAPI'.
http://labs.adobe.com/downloads/flashplayer.html

Thank you both, I will try it right now.

Are you using ppapi flash or npapi flash?

NPAPI. What is the difference, if you don't mind sharing some knowledge with a less fortunate soul

Type 'opera:flags' into Operas address bar and press enter/return. Scroll through that list for 'Tab hibernation mode' and see if it is enabled, if so, disable it.

Thanks, found it. It was already disabled. Any other ideas?

No,
I was refering to the flag:
"Tab hibernation
Mac, Windows, Linux
Enables idle background tabs to hibernate."

So I am running Windows 7 64-bit. Where might I find this setting?

If I remember correct there is a flag (option) that can be sat, that hibernates non-active tabs or "makes them less important" for the OS.
May this be related?

Are you suggesting a setting in Windows 7 that may be the culprit?

I can't figure out why this is happening. I have noticed that if I have a Flash video or game playing in the active tab and I switch to a new tab, once the focus is off of the Flash tab, it will start to slow down as if it is sharing resources with the active tab.

My PC is high-end (Core i5 4690K, 32GB DDR3, Samsung SSD, R9 280X). My hardware is not the issue.

I have updated Flash and Shokwave to the newest versions, but it keeps happening.

Any ideas?

I did have it enabled. What exactly does Turbo do that would cause this behavior?

Side note: After disabling Turbo, all is well. Thank you so much leocg.

Do you have Turbo enabled?

I do not know. I will check that and get back to you.

I have been receiving this message: http://www.pixhost.org/show/1472/23747291_untitled.jpg

Whenever I try to download from sites like these: http://www.keeplinks.eu/p/dhr4azbs

Is there a setting I can adjust to correct this in Opera 25.0?

I tried the same links in IE and they work like a charm.