Okay, I'm not sure which integrated graphics you have now. Maybe try posting that screen shot again. You could upload it to imgur and then post the link directly or press the image button on the text editor and post the link there to post the image directly.
At the point where you're at, I wouldn't be looking for a new graphics card and rather a new processor or PC altogether. I looked at benchmarks of your processor and it doesn't seem powerful enough to run dolphin.
Now that you'd have to buy a graphics card to get dolphin to run, I wouldn't bother because I doubt that you would get it to run smoothly, which makes netplay impossible due to sync issues. (small stutters are heavily amplified during netplay)
Edit: your best bet might be buying something like
this . You'd then only have to insert your old hard drive and you'll be able to use your pc like you're used to. Windows 10 will automatically adapt to the new hardware and install the necessary drivers.
You won't be able to install the ram from your previous system as it isn't compatible.