You asked (and I quote) "first I guess I am asking if Skype runs on the Android platform, if so, then presumably(?) I can connect to the internet using WiFi and use Skype as I would on my laptop .... no?"
The answer is "yes" to both. I have no idea why you have problems connecting to wifi, it could be almost anything - carrier issues, router issues, a zillion possible configuration mismatches, faulty phone...
But as to your original question, the answer is still "yes".