I'm afraid I couldn't enlighten you about the 52s origins, @CatAndHisKit as it's a while since I lived in Sheffield and it has all changed now. I used to see it while waiting for the 50 on my way home from school a long time ago
I wouldn't buy in Gleadless or Handsworth, though you'd get more for your money.
The problem with somewhere like Fraser Road is that you'd have to walk quite a long way to find anything interesting or any shops other than the usual supermarket/KFC/McD/Pets At Home combos.
I've just had a look at Rightmove. There are some decent flats in S11 (including Brincliffe, which is now pretty gentrified) - but I seem to remember Brincliffe being poor for buses (may have changed?) I think some of Brincliffe is S11 and some is S7; the S11 bit is nicer.
Only one house in S17 for that price, and it's a small renovation project in a slightly odd road that's not great for public transport (though is right on the edge of the Peak District).
The houses in S10 which fit the budget are all in Crookes. Given the combination of factors on your wish-list, I'd go for one of those. My cousin lived in Crookes for several years (she bought her first house there), and loved it. She said it was really easy to get around, and it was a short walk from her house to the bus, shops, greenery etc. It's also a pretty salubrious area - though probably best to avoid living next door to a student house...
I can't say anything about S6 as I have never lived there, and have never known anyone who does!