Okay. Finally our new connection became a reality, and I could run some tests.
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Obviously, there is a huge difference between the three - even on my old Mini. Correct me if I'm wrong, but I'm guessing that my router actually is too old, and hopefully an Extreme would give almost the same speed as directly lan through my Mini? And can the router be worse on the wi-fi-connection (pic 1) than through ethernet (pic 3), 'cause that one it handles pretty well - though not as good as lan directly to the Mini.
What type of connection is this? I think you said FIOS? What is your rated speed? Meaning, what is the internet provider committed to supply you with? 50Mb/s is nothing to sneeze at.
As far as getting an AirPort Extreme goes, you could get it and it may improve things. I don't think anyone can give you a direct yes or no answer given just the information in this thread. You seem pretty anxious about this so why not just go ahead and try it? You can always return it within Apple's return policy if it's not what you want.
Personally, I have an AirPort Time Capsule (current generation) and an older one, I think it's 2nd Gen. Yes, they last that long.