Most authorities recommend 2:1 feed at this time of year as it's nearer the concentration of honey so the bees don't have to do much work on it. A tip for mixing is to put your sugar in a plastic bucket, MARK THE LEVEL OF THE SUGAR WITH A PEN and then pour in hot tap water til it reaches the mark. As to how much to feed, this depends on how much honey the bees have stored in the brood chamber and there are formulae for how many pounds of sugar/ inches of frame you need to have but I don't have these to hand just now. I usually just feed til they won't take any more and usually find this takes 10 to 15 litres of syrup made as above. Btw this syrup is a pain to mix, it really takes ages and a lot of effort but don't be tempted to over heat or boil, as recommended in some older texts, as this causes undesirable chemical changes.
Bees do tend to like to drown in syrup, given the opportunity. A bit like "death by chocolate"? What sort of feeder are you using? It's important to make sure all parts are present and fit well to stop bees squeezing through. Don't give up on your 1:1 syrup as you'll need to make it again in spring when they need a quick burst of readily available energy if they run short before the honey flow, or you want to boost numbers.
As for hefting the hive, do so before you feed to get an idea of the empty weight then check at intervals through th winter. You'll.find you soon get an ideaof when a hive is light or heavy.