Someone else with experience with SQ can confirm some of these.
I would have thought that SQ would allow mixed class redemptions. The downside is that if you choose F on the flight to SIN and J on the flight to MXP, if it were allowed as one ticket, my assumption is that you would still be charged as F (i.e. the value of the F redemption from Australia to Europe).
If my calculations hold correct, paying F from Australia to Europe as one redemption is still cheaper in points than paying two redemptions consisting of F Australia to Singapore then J Singapore to Europe. So even if you were prepared to go only F on the first (shorter) sector to Singapore, you're still better off investigating if you can get it all onto one redemption. It may come to that you have to call the call centre and claim that it is impossible to find your redemption online (this should allow you to still access the 15% discount and avoid any assistance fees).
In any case, even with two redemptions, you should not have to recheck your bags in Singapore. Since both your flights are on SQ, they should be good enough to check through your bags at the origin port in Australia. It'd be daft silly if they could not do that.